Notes On Properties of Scalar Product - CBSE Class 12 Maths
Dot product of vectors   and $\stackrel{â†’}{\text{b}}$: $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ =  |$\stackrel{â†’}{\text{a}}$| |$\stackrel{â†’}{\text{b}}$|  cos Ï´ $\stackrel{â†’}{\text{a}}$ . ($\stackrel{â†’}{\text{b}}$ + $\stackrel{â†’}{\text{c}}$) = $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ + $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{c}}$ Distributive property of dot product over vector addition: Multiplication of dot product of vectors with a scalar: Given a scalar Î»: Î» ($\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$) = (Î» $\stackrel{â†’}{\text{a}}$). $\stackrel{â†’}{\text{b}}$ = $\stackrel{â†’}{\text{a}}$ .(Î» $\stackrel{â†’}{\text{b}}$) Let $\stackrel{â†’}{\text{a}}$ = a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$   $\stackrel{â†’}{\text{b}}$ = b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = (a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$ ).(b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$) â‡’ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = a1$\stackrel{^}{\text{i}}$  . (b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ ) +   a2 $\stackrel{^}{\text{j}}$   . (b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ ) + a3 $\stackrel{^}{\text{k}}$   . (b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ )   = a1b1 ($\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{i}}$ ) + a1b2 ($\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{j}}$ ) + a1b3 ($\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{k}}$ ) + a2b1 ($\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{i}}$ ) + a2b2 ($\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{j}}$ ) + a2b3 ($\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$ ) + a3b1 ($\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{i}}$) + a3b2 ($\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{j}}$) + a3b3 ($\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{k}}$ )  $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{j}}$  = $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$    =  $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$    = 0    $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{i}}$  = $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{j}}$    =  $\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{k}}$    = 1    â‡’ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = a1b1 (1) + a1b2 (0) + a1b3 (0) + a2b1 (0) + a2b2 (1) + a2b3 (0) + a3b1 (0) + a3b2 (0) + a3b3 (1) â‡’ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = a1b1 + a2b2  + a3b3 Projection of a Vector In âˆ† OBA: cos Î¸ = $\frac{\text{OB}}{\text{OA}}$     â‡’ OB = OA cos Î¸         â‡’ $\stackrel{â†’}{\text{OB}}$ = $\stackrel{â†’}{\text{OA}}$ cos Î¸ â‡’ $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$  cos Î¸ The magnitude of vector P is called the projection of vector A. |$\stackrel{â†’}{\text{p}}$| = |$\stackrel{â†’}{\text{a}}$| cos Ï´ If Ï´ = 0o: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ cos 0Â° $\stackrel{â†’}{\text{p}}$=$\stackrel{â†’}{\text{a}}$ (since cos 0Â° = 1) If 0o < Ï´ < 90o, then the direction of $\stackrel{â†’}{\text{p}}$ is the same as the direction of line l. $\stackrel{â†’}{\text{p}}$=$\stackrel{â†’}{\text{a}}$ cos Î¸ If Ï´ = 90o: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ cos 90Â° â‡’ $\stackrel{â†’}{\text{p}}$ =$\stackrel{â†’}{\text{O}}$ (since cos 90Â° = 0) If 90o < Ï´ < 180o, then the direction of $\stackrel{â†’}{\text{p}}$ is opposite to the direction of line l. If Ï´ = 180o: $\stackrel{â†’}{\text{p}}$ =  cos 180Â° â‡’ $\stackrel{â†’}{\text{p}}$  = - $\stackrel{â†’}{\text{a}}$ (Since cos 180Â° = -1 ) If 180o < Ï´ < 270o, then the direction of $\stackrel{â†’}{\text{p}}$ is opposite to the direction of line l. If Ï´ = 270o: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ cos 270Â° â‡’ $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{O}}$ ( Since cos 270Â° = 0) If 270o < Ï´ < 360o, then the direction of $\stackrel{â†’}{\text{p}}$ is the same as the direction of line l. Observation: If $\stackrel{^}{\text{p}}$ is a unit vector along line l: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ . $\stackrel{^}{\text{p}}$ The projection of vector $\stackrel{â†’}{\text{a}}$ along another vector $\stackrel{â†’}{\text{b}}$ is given by: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ . where is a unit vector along $\stackrel{â†’}{\text{b}}$ Since =  $\frac{\stackrel{â†’}{\text{b}}}{\text{|}\stackrel{â†’}{\text{b}}\text{|}}$     , we have: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ .  $\frac{\stackrel{â†’}{\text{b}}}{\text{|}\stackrel{â†’}{\text{b}}\text{|}}$ Or $\stackrel{â†’}{\text{p}}$ =   ($\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$) The component form of vector $\stackrel{â†’}{\text{a}}$ = a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$  : a1 = Projection of $\stackrel{â†’}{\text{a}}$ on the X-axis a2 = Projection of $\stackrel{â†’}{\text{a}}$ on the Y-axis a3 = Projection of $\stackrel{â†’}{\text{a}}$ on the Z-axis Direction cosines of vector $\stackrel{â†’}{\text{a}}$ = a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$   are given by: cos Î± = $\frac{{\text{a}}_{\text{1}}}{\left|\stackrel{â†’}{\text{a}}\right|}$  cos Î² = $\frac{{\text{a}}_{\text{2}}}{\left|\stackrel{â†’}{\text{a}}\right|}$    cos Î³ = $\frac{{\text{a}}_{\text{3}}}{\left|\stackrel{â†’}{\text{a}}\right|}$  If $\stackrel{â†’}{\text{a}}$ is a unit vector: a= cos Î±  $\stackrel{^}{\text{i}}$ + cos Î² $\stackrel{^}{\text{j}}$ + cos É£ $\stackrel{^}{\text{k}}$   :

#### Summary

Dot product of vectors   and $\stackrel{â†’}{\text{b}}$: $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ =  |$\stackrel{â†’}{\text{a}}$| |$\stackrel{â†’}{\text{b}}$|  cos Ï´ $\stackrel{â†’}{\text{a}}$ . ($\stackrel{â†’}{\text{b}}$ + $\stackrel{â†’}{\text{c}}$) = $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ + $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{c}}$ Distributive property of dot product over vector addition: Multiplication of dot product of vectors with a scalar: Given a scalar Î»: Î» ($\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$) = (Î» $\stackrel{â†’}{\text{a}}$). $\stackrel{â†’}{\text{b}}$ = $\stackrel{â†’}{\text{a}}$ .(Î» $\stackrel{â†’}{\text{b}}$) Let $\stackrel{â†’}{\text{a}}$ = a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$   $\stackrel{â†’}{\text{b}}$ = b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = (a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$ ).(b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$) â‡’ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = a1$\stackrel{^}{\text{i}}$  . (b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ ) +   a2 $\stackrel{^}{\text{j}}$   . (b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ ) + a3 $\stackrel{^}{\text{k}}$   . (b1$\stackrel{^}{\text{i}}$ + b2 $\stackrel{^}{\text{j}}$ + b3 $\stackrel{^}{\text{k}}$ )   = a1b1 ($\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{i}}$ ) + a1b2 ($\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{j}}$ ) + a1b3 ($\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{k}}$ ) + a2b1 ($\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{i}}$ ) + a2b2 ($\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{j}}$ ) + a2b3 ($\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$ ) + a3b1 ($\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{i}}$) + a3b2 ($\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{j}}$) + a3b3 ($\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{k}}$ )  $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{j}}$  = $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$    =  $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$    = 0    $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{i}}$  = $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{j}}$    =  $\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{k}}$    = 1    â‡’ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = a1b1 (1) + a1b2 (0) + a1b3 (0) + a2b1 (0) + a2b2 (1) + a2b3 (0) + a3b1 (0) + a3b2 (0) + a3b3 (1) â‡’ $\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$ = a1b1 + a2b2  + a3b3 Projection of a Vector In âˆ† OBA: cos Î¸ = $\frac{\text{OB}}{\text{OA}}$     â‡’ OB = OA cos Î¸         â‡’ $\stackrel{â†’}{\text{OB}}$ = $\stackrel{â†’}{\text{OA}}$ cos Î¸ â‡’ $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$  cos Î¸ The magnitude of vector P is called the projection of vector A. |$\stackrel{â†’}{\text{p}}$| = |$\stackrel{â†’}{\text{a}}$| cos Ï´ If Ï´ = 0o: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ cos 0Â° $\stackrel{â†’}{\text{p}}$=$\stackrel{â†’}{\text{a}}$ (since cos 0Â° = 1) If 0o < Ï´ < 90o, then the direction of $\stackrel{â†’}{\text{p}}$ is the same as the direction of line l. $\stackrel{â†’}{\text{p}}$=$\stackrel{â†’}{\text{a}}$ cos Î¸ If Ï´ = 90o: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ cos 90Â° â‡’ $\stackrel{â†’}{\text{p}}$ =$\stackrel{â†’}{\text{O}}$ (since cos 90Â° = 0) If 90o < Ï´ < 180o, then the direction of $\stackrel{â†’}{\text{p}}$ is opposite to the direction of line l. If Ï´ = 180o: $\stackrel{â†’}{\text{p}}$ =  cos 180Â° â‡’ $\stackrel{â†’}{\text{p}}$  = - $\stackrel{â†’}{\text{a}}$ (Since cos 180Â° = -1 ) If 180o < Ï´ < 270o, then the direction of $\stackrel{â†’}{\text{p}}$ is opposite to the direction of line l. If Ï´ = 270o: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ cos 270Â° â‡’ $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{O}}$ ( Since cos 270Â° = 0) If 270o < Ï´ < 360o, then the direction of $\stackrel{â†’}{\text{p}}$ is the same as the direction of line l. Observation: If $\stackrel{^}{\text{p}}$ is a unit vector along line l: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ . $\stackrel{^}{\text{p}}$ The projection of vector $\stackrel{â†’}{\text{a}}$ along another vector $\stackrel{â†’}{\text{b}}$ is given by: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ . where is a unit vector along $\stackrel{â†’}{\text{b}}$ Since =  $\frac{\stackrel{â†’}{\text{b}}}{\text{|}\stackrel{â†’}{\text{b}}\text{|}}$     , we have: $\stackrel{â†’}{\text{p}}$ = $\stackrel{â†’}{\text{a}}$ .  $\frac{\stackrel{â†’}{\text{b}}}{\text{|}\stackrel{â†’}{\text{b}}\text{|}}$ Or $\stackrel{â†’}{\text{p}}$ =   ($\stackrel{â†’}{\text{a}}$ . $\stackrel{â†’}{\text{b}}$) The component form of vector $\stackrel{â†’}{\text{a}}$ = a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$  : a1 = Projection of $\stackrel{â†’}{\text{a}}$ on the X-axis a2 = Projection of $\stackrel{â†’}{\text{a}}$ on the Y-axis a3 = Projection of $\stackrel{â†’}{\text{a}}$ on the Z-axis Direction cosines of vector $\stackrel{â†’}{\text{a}}$ = a1$\stackrel{^}{\text{i}}$ + a2 $\stackrel{^}{\text{j}}$ + a3 $\stackrel{^}{\text{k}}$   are given by: cos Î± = $\frac{{\text{a}}_{\text{1}}}{\left|\stackrel{â†’}{\text{a}}\right|}$  cos Î² = $\frac{{\text{a}}_{\text{2}}}{\left|\stackrel{â†’}{\text{a}}\right|}$    cos Î³ = $\frac{{\text{a}}_{\text{3}}}{\left|\stackrel{â†’}{\text{a}}\right|}$  If $\stackrel{â†’}{\text{a}}$ is a unit vector: a= cos Î±  $\stackrel{^}{\text{i}}$ + cos Î² $\stackrel{^}{\text{j}}$ + cos É£ $\stackrel{^}{\text{k}}$   :

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