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

#### Summary

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

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