Important Note

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Definitions and Formulas 

Scalars:- A quantity that involves only value (magnitude) or a real number.

Vectors:- A quantity that involves value (magnitude) as well as direction.

Representation:- {"id":"1-0-0-0","code":"$\\overrightarrow{\\text{AB}}$","font":{"family":"Arial","color":"#000000","size":10},"type":"$","ts":1604312965254,"cs":"Jb0DDWllWrQhefrC8Bzuig==","size":{"width":22,"height":24}} or {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-0","ts":1604317790485,"cs":"jgJyh7ByTneeNgjvPKE6sA==","size":{"width":16,"height":17}}

Here the point A from where the vector {"id":"1-1-0","code":"$\\overrightarrow{\\text{AB}}$","font":{"family":"Arial","color":"#000000","size":10},"type":"$","ts":1604312965254,"cs":"l0RNLdhyetgQI7Qr1ed8dw==","size":{"width":22,"height":24}} starts is called its initial point, and the point B where it ends is called the terminal point. The distance between the initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as{"code":"$\\left|\\overrightarrow{\\text{AB}}\\right|$","font":{"family":"Arial","size":10,"color":"#000000"},"id":"1-2-0-0","type":"$","ts":1604313768649,"cs":"c5r0C7IsKLo2uACKKH22fg==","size":{"width":33,"height":40}}.

{"code":"\\begin{align*}\n{\\overrightarrow{\\text{AB}}}&={\\left|\\overrightarrow{\\text{AB}}\\right|.\\hat{\\text{AB}}}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"2","type":"align*","ts":1604314876196,"cs":"QqL508m/DYbG3dqy5kOE/Q==","size":{"width":108,"height":40}}

Position vector:- Position vector of a point P(x,y,z) is given as {"type":"align*","code":"\\begin{align*}\n{\\overrightarrow{\\text{OP}}}&={\\overrightarrow{r}=x\\hat{i}+y\\hat{j}+z\\hat{k}}\t\n\\end{align*}","id":"39","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1604665149331,"cs":"Wsx1CJTE5Xep+iqHYuCj9A==","size":{"width":168,"height":28}} and its magnitude by {"code":"${\\sqrt[]{x^{2}+y^{2}+z^{2}}}$","font":{"color":"#000000","family":"Arial","size":10},"id":"40","type":"$","ts":1604665228727,"cs":"e4k+OAvWp9LNSJBXBbbxfA==","size":{"width":97,"height":20}}.

Direction cosines:- Consider a position vector {"font":{"size":10,"family":"Arial","color":"#000000"},"code":"$\\overrightarrow{r}$","id":"41","type":"$","ts":1604665453076,"cs":"v/SdjtqPScManbeuEIHhHA==","size":{"width":16,"height":17}} of a point P(x,y,z). Then the angles {"type":"$","font":{"color":"#000000","size":12,"family":"Arial"},"id":"42","code":"$\\alpha,\\,\\beta\\,\\,\\text{and}\\,\\,\\gamma$","ts":1604665546286,"cs":"E+tm6b3VnaBNifQeFNdqug==","size":{"width":93,"height":18}} made by the vector {"font":{"size":10,"family":"Arial","color":"#000000"},"code":"$\\overrightarrow{r}$","id":"41","type":"$","ts":1604665453076,"cs":"v/SdjtqPScManbeuEIHhHA==","size":{"width":16,"height":17}} with the positive directions of x, y, and z-axes respectively are called the direction angles. The cosine values of these angles cosα, cosβ, and cosγ are called direction cosine of the vector {"font":{"size":10,"family":"Arial","color":"#000000"},"code":"$\\overrightarrow{r}$","id":"41","type":"$","ts":1604665453076,"cs":"v/SdjtqPScManbeuEIHhHA==","size":{"width":16,"height":17}} and usually denoted by l, m, and n.

cosα = x/r ,  cosβ = y/r, and cosγ = z/r

We know that:-

l = cosα,  m= cosβ, and n= cosγ

Then lr = x, mr = y and nr = z

These numbers lr, mr, and nr are directly proportional to the direction cosines are called direction ratios of vector {"font":{"size":10,"family":"Arial","color":"#000000"},"code":"$\\overrightarrow{r}$","id":"41","type":"$","ts":1604665453076,"cs":"v/SdjtqPScManbeuEIHhHA==","size":{"width":16,"height":17}} and denoted as a, b, and c respectively.

NOTE:- l2 + m2 + n2 = 1

Components of a vector:- 


Let A(1,0,0), B(0,1,0), and C(0,0,1) respectively on the x-axis, y-axis, and z-axis

then

{"type":"align*","font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\left|\\overrightarrow{\\text{OA}}\\right|}&={1}\\\\\n{\\left|\\overrightarrow{\\text{OB}}\\right|}&={1}\\\\\n{\\left|\\overrightarrow{\\text{OC}}\\right|}&={1}\t\n\\end{align*}","id":"43","ts":1604668553467,"cs":"VBi3FS2kV6Vk6k1L4bVeeg==","size":{"width":64,"height":130}}

Each having magnitude 1 is called unit vectors along the axes OX, OY, and OZ respectively, and denoted by {"font":{"color":"#000000","size":10,"family":"Arial"},"code":"$\\hat{i}$","id":"21","type":"$","ts":1604577026579,"cs":"yxTLtdjbrEihoLpe70qRmQ==","size":{"width":8,"height":16}}, {"id":"22","font":{"size":10,"family":"Arial","color":"#000000"},"code":"$\\hat{j}$","type":"$","ts":1604577054492,"cs":"6NCfAN9NL3+PjbP8FYhaRA==","size":{"width":8,"height":18}} and {"code":"$\\hat{k}$","id":"23","font":{"color":"#000000","family":"Arial","size":10},"type":"$","ts":1604577086903,"cs":"v1MWyK4HV1VTGvS5/+x6Vg==","size":{"width":8,"height":16}} respectively.

NOTE:- (1) Two vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-1-0","ts":1604317790485,"cs":"XL6eWjYoACpF4gNgX0E1VA==","size":{"width":16,"height":17}} and {"type":"$","font":{"family":"Arial","color":"#000000","size":10},"code":"$\\overrightarrow{b}$","id":"6-2","ts":1604484903520,"cs":"B9rIRidhT9jQe1V/CdlNpQ==","size":{"width":16,"height":21}} are called collinear if and only if there exists a non-zero scalar {"id":"44","type":"$","font":{"color":"#000000","family":"Arial","size":10},"code":"$\\lambda$","ts":1604668830373,"cs":"nHx+z3fVAHVIv5E+4WiZew==","size":{"width":8,"height":10}} such that {"id":"45","type":"$","code":"$\\overrightarrow{b}=\\lambda \\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"ts":1604668864232,"cs":"Lv4V2mTJUr0jnxZ/Mt43QA==","size":{"width":61,"height":22}}. 

If the vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-1-1","ts":1604317790485,"cs":"jLe8JCL8g6hqQrAvaFTwlg==","size":{"width":16,"height":17}} and {"type":"$","font":{"family":"Arial","color":"#000000","size":10},"code":"$\\overrightarrow{b}$","id":"6-2","ts":1604484903520,"cs":"B9rIRidhT9jQe1V/CdlNpQ==","size":{"width":16,"height":21}}are given in component form then the two vectors are called collinear if and only if 

{"id":"46","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{b_{1}\\hat{i}+b_{2}\\hat{j}+b_{3}\\hat{k}}&={\\lambda\\left(a_{1}\\hat{i}+a_{2}\\hat{j}+a_{3}\\hat{k}\\right)}\\\\\n{b_{1}\\hat{i}+b_{2}\\hat{j}+b_{3}\\hat{k}}&={\\left(\\lambda a_{1}\\right)\\hat{i}+\\left(\\lambda a_{2}\\right)\\hat{j}+\\left(\\lambda a_{3}\\right)\\hat{k}}\t\n\\end{align*}","ts":1604669051318,"cs":"8vzYenxVxQhqJ/rBhZGO8g==","size":{"width":294,"height":52}}

Compare both sides:-

b1 = {"id":"44","type":"$","font":{"color":"#000000","family":"Arial","size":10},"code":"$\\lambda$","ts":1604668830373,"cs":"nHx+z3fVAHVIv5E+4WiZew==","size":{"width":8,"height":10}}a1,  b2 = {"id":"44","type":"$","font":{"color":"#000000","family":"Arial","size":10},"code":"$\\lambda$","ts":1604668830373,"cs":"nHx+z3fVAHVIv5E+4WiZew==","size":{"width":8,"height":10}}a2, and b3 = {"id":"44","type":"$","font":{"color":"#000000","family":"Arial","size":10},"code":"$\\lambda$","ts":1604668830373,"cs":"nHx+z3fVAHVIv5E+4WiZew==","size":{"width":8,"height":10}}a3

{"id":"47","type":"align*","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\frac{b_{1}}{a_{1}}}&={\\frac{b_{2}}{a_{2}}=\\frac{b_{3}}{a_{3}}=\\lambda}\t\n\\end{align*}","ts":1604669219152,"cs":"6rxmmAbO2Jg/NDZQz1rasg==","size":{"width":137,"height":34}}

(2) If {"type":"align*","code":"\\begin{align*}\n{\\overrightarrow{a}}&={a_{1}\\hat{i}+a_{2}\\hat{j}+a_{3}\\hat{k}}\t\n\\end{align*}","font":{"color":"#000000","size":10,"family":"Arial"},"id":"6-1-2","ts":1604742300712,"cs":"F/+4aeYpnP2mZvyeEfVCpg==","size":{"width":144,"height":21}} then a1, a2 and a3 are also called direction ratios of {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-1-1","ts":1604317790485,"cs":"jLe8JCL8g6hqQrAvaFTwlg==","size":{"width":16,"height":17}}.

Types of Vectors

1. Zero Vector:- A vector whose initial and terminal points coincide or same, is called a zero vector (or null vector) and denoted as {"type":"$","code":"$\\overrightarrow{\\text{O}}$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"1-3","ts":1604314206112,"cs":"aI42di3dHX/ZffLoeHmhRg==","size":{"width":16,"height":21}}or {"id":"1-0-1","code":"$\\overrightarrow{\\text{AA}}$","type":"$","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1604314357536,"cs":"+7+2gW40rjV2VtgF3cyn6w==","size":{"width":24,"height":24}} and it may be having any direction because it has no definite direction.

2. Unit Vector:- A vector whose magnitude is unity is called a unit vector and denoted as {"id":"3","code":"$\\hat{\\text{AB}}$","type":"$","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1604315046860,"cs":"99m+9Pcb/SY6UKBv2Om+kw==","size":{"width":22,"height":16}}.

For unit vector:-

{"id":"1-2-1","font":{"family":"Arial","color":"#000000","size":10},"type":"align*","code":"\\begin{align*}\n{\\left|\\overrightarrow{\\text{AB}}\\right|}&={1}\t\n\\end{align*}","ts":1604315179810,"cs":"OF+KS9MkvAjm80nBa1RzwA==","size":{"width":62,"height":40}}

3. Coinitial Vectors:- Two or more vectors having the same initial point are called coinitial vectors.

4. Collinear Vectors:- Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

5. Equal Vectors:- Two vectors are said to be equal if they have the same magnitude and direction regardless of the positions of their initial points.

If {"type":"align*","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\left|\\overrightarrow{\\text{AB}}\\right|}&={\\left|\\overrightarrow{\\text{CD}}\\right|}\t\n\\end{align*}","id":"1-2-0-1","ts":1604317595087,"cs":"5uJXyHUVg3B1UF2VvN8seA==","size":{"width":88,"height":40}} and {"font":{"size":10,"color":"#000000","family":"Arial"},"id":"4","type":"align*","code":"\\begin{align*}\n{\\hat{\\text{AB}}}&={\\hat{\\text{CD}}}\t\n\\end{align*}","ts":1604317634865,"cs":"wO7/Ig3Ru1pYGA9PSNH1CQ==","size":{"width":68,"height":20}}

Then vector is called equal vector

{"code":"\\begin{align*}\n{\\overrightarrow{\\text{AB}}}&={\\overrightarrow{\\text{CD}}}\t\n\\end{align*}","id":"5","font":{"family":"Arial","color":"#000000","size":11},"type":"align*","ts":1604317745435,"cs":"QqLVLXieNwU/kkGbhsN9wQ==","size":{"width":76,"height":32}}

6. Negative of a Vector:- A vector whose magnitude is the same as that of a given vector, but the direction is opposite to that of it, is called a negative of a given vector.

For example:- Vector {"font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\overrightarrow{\\text{BA}}$","id":"1-1-1","type":"$","ts":1604479912152,"cs":"s/Itt2ZoxsPzjdDNgle1Yw==","size":{"width":22,"height":24}} is negative of the given vector {"id":"1-1-2","code":"$\\overrightarrow{\\text{AB}}$","font":{"family":"Arial","color":"#000000","size":10},"type":"$","ts":1604312965254,"cs":"x6KWKKeducRoP7ChVjXooQ==","size":{"width":22,"height":24}}, written as 

{"id":"1-1-3","font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\overrightarrow{\\text{BA}}}&={-\\overrightarrow{\\text{AB}}}\t\n\\end{align*}","type":"align*","ts":1604479977226,"cs":"Re6eXBd3dDrtuPAUwv60hg==","size":{"width":80,"height":28}}

Example:- Find the magnitude of {"type":"align*","code":"\\begin{align*}\n{\\overrightarrow{\\text{A}}}&={3\\hat{i}+4\\hat{j}-5\\hat{k}}\t\n\\end{align*}","id":"7","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1604481280567,"cs":"59AJ7h5itocODBGSebYlyw==","size":{"width":124,"height":25}}.

Solun:- {"font":{"family":"Arial","size":10,"color":"#000000"},"id":"8","type":"align*","code":"\\begin{align*}\n{\\left|\\overrightarrow{A}\\right|}&={{\\sqrt[]{\\left(3\\right)^{2}+\\left(4\\right)^{2}+\\left(5\\right)^{2}}}}\t\n\\end{align*}","ts":1604481514980,"cs":"jBY1AI8+nbdujdUNpMLNbg==","size":{"width":184,"height":36}}

{"font":{"color":"#000000","size":10,"family":"Arial"},"code":"\\begin{align*}\n{\\left|\\overrightarrow{A}\\right|}&={{\\sqrt[]{9+16+25}}}\\\\\n{\\left|\\overrightarrow{A}\\right|}&={{\\sqrt[]{50}}}\t\n\\end{align*}","type":"align*","id":"9","ts":1604481543485,"cs":"jggNKw5ms6GmvdK9EHFKWQ==","size":{"width":141,"height":76}}

Addition of Vectors:-

{"font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\overrightarrow{\\text{AC}}}&={\\overrightarrow{\\text{AB}}+\\overrightarrow{\\text{BC}}}\t\n\\end{align*}","id":"1-1-3","type":"align*","ts":1604482062113,"cs":"13hluP+pXbZAXs1rGiYR1Q==","size":{"width":109,"height":28}}

This is known as the triangle law of addition.


Properties of vector addition:- 

1. Commutative Property:- For any two vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-1-3","ts":1604317790485,"cs":"FNTZe92yu2It5D3UGLMCFg==","size":{"width":16,"height":17}} and {"type":"$","font":{"family":"Arial","color":"#000000","size":10},"code":"$\\overrightarrow{b}$","id":"6-2","ts":1604484903520,"cs":"B9rIRidhT9jQe1V/CdlNpQ==","size":{"width":16,"height":21}}

{"font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","code":"\\begin{align*}\n{\\overrightarrow{a}+\\overrightarrow{b}}&={\\overrightarrow{b}+\\overrightarrow{a}}\t\n\\end{align*}","id":"10-0-0","ts":1604484988363,"cs":"2rhmgiVsCICqS4WgpVCvXA==","size":{"width":122,"height":25}}

2. Associative Property:- For any vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-3","ts":1604317790485,"cs":"H5bKc914QrRNIc9qQBkiZQ==","size":{"width":16,"height":17}},{"type":"$","font":{"family":"Arial","color":"#000000","size":10},"code":"$\\overrightarrow{b}$","id":"6-4","ts":1604484903520,"cs":"8kVau/RXQUoCbghl7iImBg==","size":{"width":16,"height":21}} and {"type":"$","font":{"family":"Arial","size":10,"color":"#000000"},"id":"6-5","code":"$\\overrightarrow{c}$","ts":1604485602107,"cs":"jmz8yuKpufovWTvQXTSxCA==","size":{"width":16,"height":17}}

{"id":"10-1","font":{"size":10,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\overrightarrow{a}+\\left(\\overrightarrow{b}+\\overrightarrow{c}\\right)}&={\\left(\\overrightarrow{a}+\\overrightarrow{b}\\right)+\\overrightarrow{c}}\t\n\\end{align*}","type":"align*","ts":1604485713904,"cs":"YGXMHlEcu5shE2LjRKE8lg==","size":{"width":241,"height":37}}

NOTE:- (1) For any vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6","ts":1604317790485,"cs":"eZ7SR6k5WXTeU85RpMeOaw==","size":{"width":16,"height":17}} 

{"code":"\\begin{align*}\n{\\overrightarrow{a}+\\overrightarrow{0}}&={\\overrightarrow{0}+\\overrightarrow{a}=\\overrightarrow{a}}\t\n\\end{align*}","id":"10-0-1","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"ts":1604486649751,"cs":"tQoll25WpxwLZiKgdaXgFQ==","size":{"width":160,"height":24}}

Here, the zero vector {"code":"$\\overrightarrow{0}$","font":{"color":"#000000","family":"Arial","size":10},"type":"$","id":"11-0","ts":1604486773278,"cs":"dEj/6wtc/WF40szTKEKiLQ==","size":{"width":16,"height":21}} is called the additive identity for the vector addition.

(2) For any vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6","ts":1604317790485,"cs":"eZ7SR6k5WXTeU85RpMeOaw==","size":{"width":16,"height":17}} 

{"type":"align*","id":"10-0-1","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\overrightarrow{a}+\\left(-\\overrightarrow{a}\\right)}&={\\left(-\\overrightarrow{a}\\right)+\\overrightarrow{a}=\\overrightarrow{0}}\t\n\\end{align*}","ts":1604489048549,"cs":"CSGPz5x2YEBgNJIhdZ566A==","size":{"width":222,"height":32}}

Here, the vector {"code":"$-\\overrightarrow{a}$","font":{"family":"Arial","size":10,"color":"#000000"},"id":"11-1","type":"$","ts":1604489076944,"cs":"WXNywTRnj4ZecgYsBR5M0Q==","size":{"width":28,"height":18}} is called the additive inverse or negative of vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6","ts":1604317790485,"cs":"eZ7SR6k5WXTeU85RpMeOaw==","size":{"width":16,"height":17}}.


Multiplication of a vector by a scalar:- Product of the vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-1","ts":1604317790485,"cs":"MoAZ2WP1hdOlld1qRMGy/w==","size":{"width":16,"height":17}} by the scalar {"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"12","code":"$\\lambda$","ts":1604487326128,"cs":"lkrkcyVqQQHv6Op0KRKjHw==","size":{"width":8,"height":10}}, is denoted as {"font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\lambda\\overrightarrow{a}$","id":"13-0","type":"$","ts":1604487367275,"cs":"IWew1CRF2XB5M5+kxYhIdA==","size":{"width":24,"height":17}}, is called the multiplication of a vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-0","ts":1604317790485,"cs":"ZkUtYG3GnOE42/n+Ru2UYQ==","size":{"width":16,"height":17}} by the scalar {"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"12","code":"$\\lambda$","ts":1604487326128,"cs":"lkrkcyVqQQHv6Op0KRKjHw==","size":{"width":8,"height":10}}.

Note that {"font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\lambda\\overrightarrow{a}$","id":"13-1","type":"$","ts":1604487367275,"cs":"x5MhrNCa9OGI9FGmwvKQ6A==","size":{"width":24,"height":17}} is also a vector, collinear to the vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0","ts":1604317790485,"cs":"39ZhES+vH2rf+8O11c2Tig==","size":{"width":16,"height":17}}so {"font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\lambda\\overrightarrow{a}$","id":"13","type":"$","ts":1604487367275,"cs":"d3yITc9Q+64d8KBe7RmXqA==","size":{"width":24,"height":17}} has the same direction as {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0","ts":1604317790485,"cs":"39ZhES+vH2rf+8O11c2Tig==","size":{"width":16,"height":17}}. But, the magnitude of {"font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\lambda\\overrightarrow{a}$","id":"13","type":"$","ts":1604487367275,"cs":"d3yITc9Q+64d8KBe7RmXqA==","size":{"width":24,"height":17}} is {"type":"$","font":{"color":"#000000","family":"Arial","size":10},"id":"12","code":"$\\lambda$","ts":1604487326128,"cs":"lkrkcyVqQQHv6Op0KRKjHw==","size":{"width":8,"height":10}} times the magnitude of vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0","ts":1604317790485,"cs":"39ZhES+vH2rf+8O11c2Tig==","size":{"width":16,"height":17}}.

{"code":"\\begin{align*}\n{\\left|\\lambda \\overrightarrow{a}\\right|}&={\\left|\\lambda\\right|\\left|\\overrightarrow{a}\\right|}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"id":"14","ts":1604488092201,"cs":"B0f16oEK3ybDZb7g2inZxA==","size":{"width":100,"height":28}}

Product of two vectors:- Multiplication of two vectors is also defined in two ways- scalar product and vector product. Scalar product where the result is scalar and vector product where the result is a vector.

1. Scalar (or dot) product of two vectors:-

The scalar product of two non-zero vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-0","ts":1604317790485,"cs":"jzD3YNPJrJh8zW5pzTYtEw==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-0","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"K5+gMMPhfm9jPRvjTgXDHA==","size":{"width":16,"height":21}}, denoted by {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2","ts":1604317790485,"cs":"zeCo+KBRxW5L9t5EszQ3Bw==","size":{"width":16,"height":17}}.{"id":"6-0-2-2-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"lyPY/ZfLMzrOlMO0jaT8jA==","size":{"width":16,"height":21}} is defined as

{"font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","id":"15-0-0","code":"\\begin{align*}\n{\\overrightarrow{a}.\\overrightarrow{\\text{b}}}&={\\left|\\overrightarrow{a}\\right|\\left|\\overrightarrow{\\text{b}}\\right|\\cos\\theta}\t\n\\end{align*}","ts":1604573807392,"cs":"G0t1coJX2sdB889jjzeJyA==","size":{"width":146,"height":36}}

where θ is the angle between {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-1-0","ts":1604317790485,"cs":"9SYsvws5hkzUEU2a1xFh3w==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"cxzNvVL+FKiJ85b3l+b4gA==","size":{"width":16,"height":21}}, {"code":"\\begin{align*}\n{0}&\\leqslant{\\theta\\leqslant\\Pi}\t\n\\end{align*}","id":"16","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","ts":1604575027091,"cs":"hImLgaYE0d96NED9YPja6Q==","size":{"width":68,"height":16}}.

Properties of scalar product:-

1. The scalar product is commutative i.e.

{"type":"align*","id":"15-1-0-0-0-0","code":"\\begin{align*}\n{\\overrightarrow{a}.\\overrightarrow{\\text{b}}}&={\\overrightarrow{\\text{b}}.\\overrightarrow{a}}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"ts":1604575390957,"cs":"FvdbtXKOe2jVWxqsUrruxA==","size":{"width":98,"height":25}}

2. Distributivity of scalar product over addition:- Let {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"19","ts":1604317790485,"cs":"pSP6cCIGcv2mjf54DpfmWw==","size":{"width":16,"height":17}},{"id":"17","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"0XuXFL/VIby9SxNJn5xOvA==","size":{"width":16,"height":21}}and{"font":{"color":"#000000","size":10,"family":"Arial"},"id":"18","code":"$\\overrightarrow{\\text{c}}$","type":"$","ts":1604575639939,"cs":"qfmdIm0vuXQTLmYqIZb0tQ==","size":{"width":16,"height":17}}be any three vectors, then

{"code":"\\begin{align*}\n{\\overrightarrow{a}.\\left(\\overrightarrow{\\text{b}}+\\overrightarrow{c}\\right)}&={\\overrightarrow{a}.\\overrightarrow{\\text{b}}+\\overrightarrow{a}.\\overrightarrow{c}}\t\n\\end{align*}","type":"align*","id":"15-1-1-0","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1604575785836,"cs":"8sVm7Zd76wxDXdV7xuRzEg==","size":{"width":216,"height":37}}

3. Let {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-1","ts":1604317790485,"cs":"FO3DzyaedHmXfz2Y8R038w==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"vxHK6r82gKcvo+pCv5wT+A==","size":{"width":16,"height":21}} be any two vectors, λ be any scalars. Then

{"code":"\\begin{align*}\n{\\left(\\lambda\\overrightarrow{a}\\right).\\overrightarrow{\\text{b}}}&={\\lambda\\left(\\overrightarrow{a}.\\overrightarrow{\\text{b}}\\right)=\\overrightarrow{a}.\\left(\\lambda\\overrightarrow{\\text{b}}\\right)}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","id":"15-1-0-1","ts":1604576174753,"cs":"qkj2aIO80vrwyZbk7H+eSg==","size":{"width":252,"height":37}}

4. Let {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-2","ts":1604317790485,"cs":"4/LzPVEQSnn01qH6NN+J3A==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-2","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"7CCaMNScJHx79JHX+OhRtg==","size":{"width":16,"height":21}} be two non-zero vectors, then 

(i) {"code":"\\begin{align*}\n{\\overrightarrow{a}.\\overrightarrow{\\text{b}}}&={0}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"15-1-0-0-1-0","type":"align*","ts":1604576340751,"cs":"J+UKtdhRiXgY9+CuSm55qw==","size":{"width":68,"height":25}} if and only if both the vectors are perpendicular to each other or θ = 90o.

(ii) If θ = 0 or both the vectors are parallel to each other then 

{"type":"align*","id":"15-0-1-0-0-0","code":"\\begin{align*}\n{\\overrightarrow{a}.\\overrightarrow{\\text{b}}}&={\\left|\\overrightarrow{a}\\right|\\left|\\overrightarrow{\\text{b}}\\right|}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"ts":1604576524565,"cs":"ODSSp3WfHPwT+kpBsu/wgg==","size":{"width":112,"height":36}}

NOTE:- {"font":{"size":10,"color":"#000000","family":"Arial"},"id":"15-0-1-1","code":"\\begin{align*}\n{\\overrightarrow{a}.\\overrightarrow{a}}&={\\left|\\overrightarrow{a}\\right|^{2}}\t\n\\end{align*}","type":"align*","ts":1604576638242,"cs":"CwiXT8iKu7ObAHHzRD/yqA==","size":{"width":92,"height":30}}

(ii) If θ = Ï€ or both the vectors are parallel to each other and direction is opposite to each other then 

{"code":"\\begin{align*}\n{\\overrightarrow{a}.\\overrightarrow{\\text{b}}}&={-\\left|\\overrightarrow{a}\\right|\\left|\\overrightarrow{\\text{b}}\\right|}\t\n\\end{align*}","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","id":"15-0-1-0-1","ts":1604576747206,"cs":"ByWtr3FZe9fBD8yMQubk6g==","size":{"width":124,"height":36}}

5. The angle between two non-zero vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-3","ts":1604317790485,"cs":"Z1g9fpp0BWN4irPn8OPoJg==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-3-0","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"1KC7BFq/R1ommPlAphoHyA==","size":{"width":16,"height":21}} is

{"code":"\\begin{align*}\n{\\cos\\theta}&={\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\left|\\overrightarrow{a}\\right|.\\left|\\overrightarrow{b}\\right|}}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"id":"20-0","type":"align*","ts":1604576883818,"cs":"6lPMF0eBkw5yFBcI9JE7Jw==","size":{"width":118,"height":68}}

NOTE:- As conventionally {"font":{"color":"#000000","size":10,"family":"Arial"},"code":"$\\hat{i}$","id":"21","type":"$","ts":1604577026579,"cs":"yxTLtdjbrEihoLpe70qRmQ==","size":{"width":8,"height":16}}, {"id":"22","font":{"size":10,"family":"Arial","color":"#000000"},"code":"$\\hat{j}$","type":"$","ts":1604577054492,"cs":"6NCfAN9NL3+PjbP8FYhaRA==","size":{"width":8,"height":18}} and {"code":"$\\hat{k}$","id":"23","font":{"color":"#000000","family":"Arial","size":10},"type":"$","ts":1604577086903,"cs":"v1MWyK4HV1VTGvS5/+x6Vg==","size":{"width":8,"height":16}} are mutually perpendicular unit vectors then

{"id":"24-0","code":"\\begin{align*}\n{\\hat{i}.\\hat{i}}&={\\hat{j}.\\hat{j}=\\hat{k}.\\hat{k}=1}\\\\\n{\\hat{i}.\\hat{j}}&={\\hat{j}.\\hat{k}=\\hat{k}.\\hat{i}=0}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"ts":1604577205531,"cs":"kZJoGUViJtPvn7dj9Wg7pQ==","size":{"width":140,"height":44}}

Projection of a vector on a line:-

(1) Projection of vector {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-4-0","ts":1604317790485,"cs":"GKeKi4kZ5vQDnRJz8ePvYQ==","size":{"width":16,"height":17}} on {"id":"6-0-2-2-0-0-4","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"IsmDzGGEsDH3aThtgD3GRw==","size":{"width":16,"height":21}}is

{"id":"25-0","font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{gather*}\n{=\\left(\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\left|\\overrightarrow{b}\\right|}\\right)}\\\\\n{or}\\\\\n{=\\left|a\\right|\\cos\\theta}\\\\\n{or}\\\\\n{=\\overrightarrow{a}.\\hat{b}}\t\n\\end{gather*}","type":"gather*","ts":1604581122917,"cs":"zIFeM8eXQO4XC0rJ4Vz/WQ==","size":{"width":89,"height":165}}

And {"id":"26-0","code":"$\\overrightarrow{a}.\\hat{b}$","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1604581178627,"cs":"lFCbgZ9Ei7MyncnUXjMGtg==","size":{"width":30,"height":17}} denotes the projection of {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"29","ts":1604317790485,"cs":"V74kmNGZNVPXnvy4ElJuyA==","size":{"width":16,"height":17}} in the direction of {"id":"27","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"HvD9insdEKUk6O3RlQ+oUA==","size":{"width":16,"height":21}}.

(2) Projection of vector {"id":"6-0-2-2-0-0-4","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"IsmDzGGEsDH3aThtgD3GRw==","size":{"width":16,"height":21}}on{"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-4","ts":1604317790485,"cs":"Ub6CViUjFBljZFlGis0Yag==","size":{"width":16,"height":17}}is

{"code":"\\begin{gather*}\n{=\\left(\\frac{\\overrightarrow{a}.\\overrightarrow{b}}{\\left|\\overrightarrow{a}\\right|}\\right)}\\\\\n{or}\\\\\n{=\\left|b\\right|\\cos\\theta}\\\\\n{or}\\\\\n{=\\overrightarrow{\\text{b}}.\\hat{a}}\t\n\\end{gather*}","type":"gather*","id":"25-1","font":{"size":10,"color":"#000000","family":"Arial"},"ts":1604581353257,"cs":"P8JB07s8D0od67kwVeZ9Xw==","size":{"width":89,"height":153}}


And {"id":"26-1-0","font":{"color":"#000000","size":10,"family":"Arial"},"code":"$\\overrightarrow{\\text{b}}.\\hat{a}$","type":"$","ts":1604581392671,"cs":"myFu4j78zwMGahuHNNMUMg==","size":{"width":30,"height":21}} denotes the projection of {"id":"31","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"TowMsDpVqt1dPnKyu/vyLw==","size":{"width":16,"height":21}} in the direction of {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0","ts":1604317790485,"cs":"VZaz1SLQF4QUltCXu6aGbA==","size":{"width":16,"height":17}}.

NOTE:- If {"id":"32","font":{"color":"#000000","size":10,"family":"Arial"},"type":"$","code":"$\\hat{\\text{p}}$","ts":1604581602598,"cs":"zyw3QJ7Fms1RRsFtUG7dxw==","size":{"width":8,"height":14}} is the unit vector along line l then the projection of{"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0","ts":1604317790485,"cs":"VZaz1SLQF4QUltCXu6aGbA==","size":{"width":16,"height":17}} on the line is given by:-

{"code":"$\\overrightarrow{\\text{a}}.\\hat{\\text{p}}$","font":{"size":10,"color":"#000000","family":"Arial"},"id":"26-1-1","type":"$","ts":1604581817280,"cs":"4JAVAB7suGh+40WDZW/i9g==","size":{"width":32,"height":20}}.

2. Vector (or cross) product of two vectors:-

The vector product of two non-zero vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-0","ts":1604317790485,"cs":"jzD3YNPJrJh8zW5pzTYtEw==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-0","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"K5+gMMPhfm9jPRvjTgXDHA==","size":{"width":16,"height":21}}, denoted by {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2","ts":1604317790485,"cs":"zeCo+KBRxW5L9t5EszQ3Bw==","size":{"width":16,"height":17}}x{"id":"6-0-2-2-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"lyPY/ZfLMzrOlMO0jaT8jA==","size":{"width":16,"height":21}} is defined as

{"type":"align*","code":"\\begin{align*}\n{\\overrightarrow{a}\\times\\overrightarrow{\\text{b}}}&={\\left|\\overrightarrow{a}\\right|\\left|\\overrightarrow{\\text{b}}\\right|\\sin\\theta.\\hat{n}}\t\n\\end{align*}","id":"15-0-0","font":{"color":"#000000","size":10,"family":"Arial"},"ts":1604650886702,"cs":"qJoz+rqSWF3TzJR/OZtOag==","size":{"width":173,"height":36}}

where θ is the angle between {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-1-1","ts":1604317790485,"cs":"GuhGc2xlhKWsVD6S4k6tVA==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"cxzNvVL+FKiJ85b3l+b4gA==","size":{"width":16,"height":21}}, {"code":"\\begin{align*}\n{0}&\\leqslant{\\theta\\leqslant\\Pi}\t\n\\end{align*}","id":"16","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","ts":1604575027091,"cs":"hImLgaYE0d96NED9YPja6Q==","size":{"width":68,"height":16}} and {"code":"$\\hat{n}$","type":"$","font":{"size":10,"family":"Arial","color":"#000000"},"id":"33","ts":1604650915198,"cs":"wWfg9TqmtoYC3voN/MF9+A==","size":{"width":8,"height":12}} is

a unit vector perpendicular to both {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-1-2","ts":1604317790485,"cs":"ZDHiaLe8Ea2cRnmy9CTHyA==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"cxzNvVL+FKiJ85b3l+b4gA==","size":{"width":16,"height":21}}.

{"type":"align*","id":"53","code":"\\begin{align*}\n{\\overrightarrow{a}\\times \\overrightarrow{b}}&={\\begin{vmatrix}\n{\\hat{i}}&{\\hat{j}}&{\\hat{k}}\\\\\n{a_{1}}&{a_{2}}&{a_{3}}\\\\\n{b_{1}}&{b_{2}}&{b_{3}}\\\\\n\\end{vmatrix}}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"ts":1604751692844,"cs":"EiFlqGUJwWpyjZ/7jTIRQQ==","size":{"width":158,"height":64}}

Properties of vector product:-

1. The scalar product is not commutative i.e.

{"font":{"color":"#000000","size":10,"family":"Arial"},"type":"align*","id":"15-1-0-0-0-1","code":"\\begin{align*}\n{\\overrightarrow{a}\\times\\overrightarrow{\\text{b}}}&\\neq{\\overrightarrow{\\text{b}}\\times\\overrightarrow{a}}\\\\\n{\\overrightarrow{a}\\times\\overrightarrow{\\text{b}}}&={-\\overrightarrow{\\text{b}}\\times\\overrightarrow{a}}\t\n\\end{align*}","ts":1604651269979,"cs":"ktu+0YvI7PLGxtyMIxqNXA==","size":{"width":136,"height":56}}

Because in vector product direction of the resultant vector differs from the initial direction.

2. Distributivity of vector product over addition:- Let {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"19","ts":1604317790485,"cs":"pSP6cCIGcv2mjf54DpfmWw==","size":{"width":16,"height":17}},{"id":"17","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"0XuXFL/VIby9SxNJn5xOvA==","size":{"width":16,"height":21}}and{"font":{"color":"#000000","size":10,"family":"Arial"},"id":"18","code":"$\\overrightarrow{\\text{c}}$","type":"$","ts":1604575639939,"cs":"qfmdIm0vuXQTLmYqIZb0tQ==","size":{"width":16,"height":17}}be any three vectors, then

{"code":"\\begin{align*}\n{\\overrightarrow{a}\\times\\left(\\overrightarrow{\\text{b}}+\\overrightarrow{c}\\right)}&={\\overrightarrow{a}\\times\\overrightarrow{\\text{b}}+\\overrightarrow{a}\\times\\overrightarrow{c}}\t\n\\end{align*}","type":"align*","font":{"color":"#000000","size":10,"family":"Arial"},"id":"15-1-1-1-0","ts":1604651933764,"cs":"tYqhOibHuEdw0YPOfwp00A==","size":{"width":253,"height":37}}

3. Let {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-1","ts":1604317790485,"cs":"FO3DzyaedHmXfz2Y8R038w==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"vxHK6r82gKcvo+pCv5wT+A==","size":{"width":16,"height":21}} be any two vectors, λ be any scalars. Then

{"type":"align*","font":{"color":"#000000","family":"Arial","size":10},"id":"34","code":"\\begin{align*}\n{\\lambda\\left(\\overrightarrow{a}\\times\\overrightarrow{\\text{b}}\\right)}&={\\left(\\lambda\\overrightarrow{a}\\right)\\times\\overrightarrow{\\text{b}}=\\overrightarrow{a}\\times\\left(\\lambda\\overrightarrow{\\text{b}}\\right)}\t\n\\end{align*}","ts":1604651905003,"cs":"Xr0yHXAbSQ/MYB8ZFT1R8Q==","size":{"width":288,"height":37}}

3. Let {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-2","ts":1604317790485,"cs":"4/LzPVEQSnn01qH6NN+J3A==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-2","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"7CCaMNScJHx79JHX+OhRtg==","size":{"width":16,"height":21}} be two non-zero vectors, then 

(i) {"code":"\\begin{align*}\n{\\overrightarrow{a}\\times\\overrightarrow{\\text{b}}}&={0}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","color":"#000000","size":10},"id":"15-1-0-0-1-1","ts":1604652021245,"cs":"XBbZfrzDQ604naOeAErQQA==","size":{"width":80,"height":25}} if both the vectors are parallel or collinear or θ = 0o.

(ii) If θ = 90o or both the vectors are perpendicular to each other then 

{"code":"\\begin{align*}\n{\\overrightarrow{a}\\times\\overrightarrow{\\text{b}}}&={\\left|\\overrightarrow{a}\\right|\\left|\\overrightarrow{\\text{b}}\\right|}\t\n\\end{align*}","font":{"size":10,"color":"#000000","family":"Arial"},"type":"align*","id":"15-0-1-0-0-1","ts":1604652067334,"cs":"0e6PDGzIHz+FbxfuWVxOLw==","size":{"width":124,"height":36}}

4. The angle between two non-zero vectors {"type":"$","code":"$\\overrightarrow{a}$","font":{"color":"#000000","family":"Arial","size":10},"id":"6-0-2-1-0-3","ts":1604317790485,"cs":"Z1g9fpp0BWN4irPn8OPoJg==","size":{"width":16,"height":17}} and {"id":"6-0-2-2-0-0-3-1","type":"$","font":{"size":10,"color":"#000000","family":"Arial"},"code":"$\\overrightarrow{\\text{b}}$","ts":1604573669560,"cs":"YtwUGv/LtMNZCkLMgWvIVA==","size":{"width":16,"height":21}} is

{"font":{"size":10,"family":"Arial","color":"#000000"},"id":"20-1","type":"align*","code":"\\begin{align*}\n{\\sin\\theta}&={\\frac{\\overrightarrow{a}\\times\\overrightarrow{b}}{\\left|\\overrightarrow{a}\\right|.\\left|\\overrightarrow{b}\\right|}}\t\n\\end{align*}","ts":1604652250248,"cs":"/kXxCXQYamdjCvLd97261A==","size":{"width":116,"height":68}}

NOTE:- As conventionally {"font":{"color":"#000000","size":10,"family":"Arial"},"code":"$\\hat{i}$","id":"21","type":"$","ts":1604577026579,"cs":"yxTLtdjbrEihoLpe70qRmQ==","size":{"width":8,"height":16}}, {"id":"22","font":{"size":10,"family":"Arial","color":"#000000"},"code":"$\\hat{j}$","type":"$","ts":1604577054492,"cs":"6NCfAN9NL3+PjbP8FYhaRA==","size":{"width":8,"height":18}} and {"code":"$\\hat{k}$","id":"23","font":{"color":"#000000","family":"Arial","size":10},"type":"$","ts":1604577086903,"cs":"v1MWyK4HV1VTGvS5/+x6Vg==","size":{"width":8,"height":16}} are mutually perpendicular unit vectors then

{"id":"24-1","font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\hat{i}\\times\\hat{i}}&={\\hat{j}\\times\\hat{j}=\\hat{k}\\times\\hat{k}=0}\t\n\\end{align*}","type":"align*","ts":1604652347899,"cs":"c+omsxOB0ZMXPGZQtUIIxg==","size":{"width":177,"height":20}}

{"code":"\\begin{align*}\n{\\hat{i}\\times\\hat{j}}&={\\hat{k}}\\\\\n{\\hat{j}\\times\\hat{k}}&={\\hat{i}}\\\\\n{\\hat{k}\\times\\hat{i}}&={\\hat{j}}\t\n\\end{align*}","font":{"size":10,"family":"Arial","color":"#000000"},"id":"35","type":"align*","ts":1604652423298,"cs":"bH46BUKPqYr3dz71e5b8VQ==","size":{"width":64,"height":69}}


5. Area of Triangle:-

{"code":"\\begin{align*}\n{\\text{area}\\left(\\Delta \\text{ABC}\\right)}&={\\frac{1}{2}\\left|\\overrightarrow{a}\\times \\overrightarrow{b}\\right|}\t\n\\end{align*}","type":"align*","font":{"family":"Arial","size":11,"color":"#000000"},"id":"36-0","ts":1604652605812,"cs":"AkP15mIyY2NM2fw2M7M2pA==","size":{"width":212,"height":40}}

Proof:-

{"type":"align*","id":"37","font":{"family":"Arial","color":"#000000","size":10},"code":"\\begin{align*}\n{\\text{area}\\left(\\Delta \\text{ABC}\\right)}&={\\frac{1}{2}\\times \\text{Base}\\times \\text{Height}}\\\\\n{\\text{area}\\left(\\Delta \\text{ABC}\\right)}&={\\frac{1}{2}\\times \\left|\\overrightarrow{b}\\right|\\times \\text{|}\\overrightarrow{a}\\text{|}\\sin\\theta}\\\\\n{\\text{area}\\left(\\Delta \\text{ABC}\\right)}&={\\frac{1}{2}\\times \\left(\\overrightarrow{a}\\times\\overrightarrow{b}\\right)}\t\n\\end{align*}","ts":1604655852163,"cs":"TwjQ5RJ/s9/IlknprAGJJA==","size":{"width":246,"height":114}}

5. Area of Parallelogram:-

{"font":{"size":11,"family":"Arial","color":"#000000"},"code":"\\begin{align*}\n{\\text{area}}&={\\left|\\overrightarrow{a}\\times \\overrightarrow{b}\\right|}\t\n\\end{align*}","type":"align*","id":"36-1-0","ts":1604662188158,"cs":"EP3EtvOnItJAJpus4J0unw==","size":{"width":128,"height":40}}

Proof:-

{"type":"align*","id":"37","code":"\\begin{align*}\n{\\text{area}\\left(\\Delta \\text{ABC}\\right)}&={\\text{Base}\\times \\text{Height}}\\\\\n{\\text{area}\\left(\\Delta \\text{ABC}\\right)}&={\\left|\\overrightarrow{b}\\right|\\times \\text{|}\\overrightarrow{a}\\text{|}\\sin\\theta}\\\\\n{\\text{area}\\left(\\Delta \\text{ABC}\\right)}&={\\overrightarrow{a}\\times\\overrightarrow{b}}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"ts":1604663158601,"cs":"ZiPOdQMpJxI/BewARk0WCw==","size":{"width":212,"height":86}}

6. If three points are collinear then

{"id":"38-0","font":{"size":10,"family":"Arial","color":"#000000"},"type":"align*","code":"\\begin{align*}\n{\\overrightarrow{a}\\times \\overrightarrow{b}+\\overrightarrow{b}\\times \\overrightarrow{c}+\\overrightarrow{c}\\times\\overrightarrow{a}}&={0}\t\n\\end{align*}","ts":1604663686563,"cs":"1QJcw2KycIlt/V7Bc0IcOg==","size":{"width":222,"height":25}}

7. Find a unit vector

{"code":"\\begin{align*}\n{\\hat{n}}&={\\frac{\\overrightarrow{a}\\times \\overrightarrow{b}}{\\left|\\overrightarrow{a}\\right|\\left|\\overrightarrow{b}\\right|\\sin\\theta}}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"id":"38-1","type":"align*","ts":1604663815067,"cs":"iQek4NCukzZuARC2kp63Pg==","size":{"width":122,"height":68}}

Vector joining two points:-

If P1(x1,y1,z1) and P2(x2,y2,z2) are two points, then the vector joining P1 and P2 is the vector {"font":{"family":"Arial","size":10,"color":"#000000"},"code":"$\\overrightarrow{P_{1}P_{2}}$","type":"$","id":"48","ts":1604742695732,"cs":"vxkL5Q8zhlbD1rVLkwzW7w==","size":{"width":32,"height":25}}.


According to the triangle law of addition:-

{"id":"49","type":"align*","code":"\\begin{align*}\n{\\overrightarrow{\\text{OP}_{1}}+\\overrightarrow{\\text{P}_{1}\\text{P}_{2}}}&={\\overrightarrow{\\text{OP}_{2}}}\t\n\\end{align*}","font":{"family":"Arial","color":"#000000","size":10},"ts":1604743894642,"cs":"F3VT15lCLKhVMmCG+R1gzw==","size":{"width":133,"height":28}}

{"id":"50","type":"align*","font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\overrightarrow{\\text{P}_{1}\\text{P}_{2}}}&={\\overrightarrow{\\text{OP}_{2}}-\\overrightarrow{\\text{OP}_{1}}}\\\\\n{\\overrightarrow{\\text{P}_{1}\\text{P}_{2}}}&={\\left(x_{2}\\hat{i}+y_{2}\\hat{j}+z_{2}\\hat{k}\\right)-\\left(x_{1}\\hat{i}+y_{1}\\hat{\\hat{j}}+z_{1}\\hat{\\hat{k}}\\right)}\\\\\n{\\overrightarrow{\\text{P}_{1}\\text{P}_{2}}}&={\\left(x_{2}-x_{1}\\right)\\hat{i}+\\left(y_{2}-y_{1}\\right)\\hat{\\hat{j}}+\\left(z_{2}-z_{1}\\right)\\hat{k}}\\\\\n{\\overrightarrow{\\text{P}_{1}\\text{P}_{2}}}&={\\left(x_{2}-x_{1}\\right)\\hat{i}+\\left(y_{2}-y_{1}\\right)\\hat{j}+\\left(z_{2}-z_{1}\\right)\\hat{k}}\t\n\\end{align*}","ts":1604744250059,"cs":"PZ9RaGdgPnNFAKEvsS8saQ==","size":{"width":328,"height":136}}

{"font":{"size":10,"color":"#000000","family":"Arial"},"code":"\\begin{align*}\n{\\left|\\overrightarrow{\\text{P}_{1}\\text{P}_{2}}\\right|}&={{\\sqrt[]{\\left(x_{2}-x_{1}\\right)^{2}+\\left(y_{2}-y_{1}\\right)^{2}+\\left(z_{2}-z_{1}\\right)^{2}}}}\t\n\\end{align*}","id":"51","type":"align*","ts":1604744319015,"cs":"lCmJYLZfnNYcOVpQRixfkQ==","size":{"width":321,"height":40}}

Section Formula:-

Case 1: When R divides PQ internally 

{"id":"52","type":"align*","code":"\\begin{align*}\n{\\overrightarrow{r}}&={\\frac{m\\overrightarrow{b}+n\\overrightarrow{a}}{m+n}}\t\n\\end{align*}","font":{"color":"#000000","family":"Arial","size":10},"ts":1604746272477,"cs":"B8k5o0IneSpOfTnGjmCsLg==","size":{"width":118,"height":44}}

{"font":{"family":"Arial","size":10,"color":"#000000"},"code":"\\begin{align*}\n{\\overrightarrow{\\text{OR}}}&={\\frac{m\\overrightarrow{b}+n\\overrightarrow{a}}{m+n}}\t\n\\end{align*}","type":"align*","id":"52","ts":1604746253516,"cs":"GqB18AWEn0bbaPuWR+OHhA==","size":{"width":126,"height":44}}

Case 2: When R divides PQ externally 

{"font":{"color":"#000000","family":"Arial","size":10},"code":"\\begin{align*}\n{\\overrightarrow{r}}&={\\frac{m\\overrightarrow{b}-n\\overrightarrow{a}}{m-n}}\t\n\\end{align*}","type":"align*","id":"52","ts":1604750769982,"cs":"uTcQ2etESkHduxt57+IEJQ==","size":{"width":118,"height":44}}

{"code":"\\begin{align*}\n{\\overrightarrow{\\text{OR}}}&={\\frac{m\\overrightarrow{b}-n\\overrightarrow{a}}{m-n}}\t\n\\end{align*}","id":"52","font":{"family":"Arial","size":10,"color":"#000000"},"type":"align*","ts":1604750785648,"cs":"FOgVHjcHKEpO+/9SDctKDw==","size":{"width":126,"height":44}}

NOTE:- If R is the midpoint of PQ then m = n

{"font":{"family":"Arial","color":"#000000","size":10},"type":"align*","code":"\\begin{align*}\n{\\overrightarrow{\\text{OR}}}&={\\frac{\\overrightarrow{a}+\\overrightarrow{b}}{2}}\t\n\\end{align*}","id":"52","ts":1604746643670,"cs":"tWOVVIEdgcOYfVJnlFgdNQ==","size":{"width":102,"height":42}}


Download PDF notes of Vector Algebra


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