Notes On Scalar Product - CBSE Class 12 Maths
Vector multiplication is of two types. One is dot or scalar product and the other is cross or vector product. The name itself indicates the result. If we use scalar product, the result is a scalar, that is, a real number. If we use vector product, the result is a vector. Dot product of vectors $\stackrel{\to }{\text{a}}$ and $\stackrel{\to }{\text{b}}$: $\stackrel{\to }{\text{a}}$ . $\stackrel{\to }{\text{b}}$ = |$\stackrel{\to }{\text{a}}$| || cos ϴ, where 0o ≤ ϴ ≤ 180o If $\stackrel{\to }{\text{a}}$ = $\stackrel{\to }{\text{0}}$ or = $\stackrel{\to }{\text{0}}$ $\stackrel{\to }{\text{a}}$ . $\stackrel{\to }{\text{b}}$ = 0 Properties of dot product: (1) $\stackrel{\to }{\text{a}}$ . $\stackrel{\to }{\text{b}}$  ∈  R  (2) If $\stackrel{\to }{\text{a}}$  . $\stackrel{\to }{\text{b}}$  = 0 ⇔  $\stackrel{\to }{\text{a}}$  ⊥  $\stackrel{\to }{\text{b}}$   (3) When θ = 0° , $\stackrel{\to }{\text{a}}$   .  = |$\stackrel{\to }{\text{a}}$| ||        (i) $\stackrel{\to }{\text{a}}$   . $\stackrel{\to }{\text{a}}$   = |$\stackrel{\to }{\text{a}}$|2 (4) When θ = 180°,  $\stackrel{\to }{\text{a}}$   .  = - |$\stackrel{\to }{\text{a}}$| || (5) If θ is the angle between the vectors  $\stackrel{\to }{\text{a}}$   and  $\stackrel{\to }{\text{b}}$$\stackrel{}{\text{}}$$\stackrel{}{\text{}}$            Cos θ  = $\frac{\text{}\stackrel{\to }{\text{a}}\text{.}\stackrel{\to }{\text{b}}}{\text{|}\stackrel{_}{\text{a}}\text{| |}\stackrel{_}{\text{b}}\text{|}}$ For unit vectors $\stackrel{^}{\text{i}}$  , $\stackrel{^}{\text{j}}$ and : $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{i}}$   = |$\stackrel{^}{\text{i}}$|2  = 1 $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{j}}$   = ||2  = 1 $\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{k}}$   = ||2  = 1 For mutually perpendicular unit vectors  $\stackrel{^}{\text{i}}$  , $\stackrel{^}{\text{j}}$ and : $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{j}}$   = 0 $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$   = 0 $\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{i}}$   = 0

#### Summary

Vector multiplication is of two types. One is dot or scalar product and the other is cross or vector product. The name itself indicates the result. If we use scalar product, the result is a scalar, that is, a real number. If we use vector product, the result is a vector. Dot product of vectors $\stackrel{\to }{\text{a}}$ and $\stackrel{\to }{\text{b}}$: $\stackrel{\to }{\text{a}}$ . $\stackrel{\to }{\text{b}}$ = |$\stackrel{\to }{\text{a}}$| || cos ϴ, where 0o ≤ ϴ ≤ 180o If $\stackrel{\to }{\text{a}}$ = $\stackrel{\to }{\text{0}}$ or = $\stackrel{\to }{\text{0}}$ $\stackrel{\to }{\text{a}}$ . $\stackrel{\to }{\text{b}}$ = 0 Properties of dot product: (1) $\stackrel{\to }{\text{a}}$ . $\stackrel{\to }{\text{b}}$  ∈  R  (2) If $\stackrel{\to }{\text{a}}$  . $\stackrel{\to }{\text{b}}$  = 0 ⇔  $\stackrel{\to }{\text{a}}$  ⊥  $\stackrel{\to }{\text{b}}$   (3) When θ = 0° , $\stackrel{\to }{\text{a}}$   .  = |$\stackrel{\to }{\text{a}}$| ||        (i) $\stackrel{\to }{\text{a}}$   . $\stackrel{\to }{\text{a}}$   = |$\stackrel{\to }{\text{a}}$|2 (4) When θ = 180°,  $\stackrel{\to }{\text{a}}$   .  = - |$\stackrel{\to }{\text{a}}$| || (5) If θ is the angle between the vectors  $\stackrel{\to }{\text{a}}$   and  $\stackrel{\to }{\text{b}}$$\stackrel{}{\text{}}$$\stackrel{}{\text{}}$            Cos θ  = $\frac{\text{}\stackrel{\to }{\text{a}}\text{.}\stackrel{\to }{\text{b}}}{\text{|}\stackrel{_}{\text{a}}\text{| |}\stackrel{_}{\text{b}}\text{|}}$ For unit vectors $\stackrel{^}{\text{i}}$  , $\stackrel{^}{\text{j}}$ and : $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{i}}$   = |$\stackrel{^}{\text{i}}$|2  = 1 $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{j}}$   = ||2  = 1 $\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{k}}$   = ||2  = 1 For mutually perpendicular unit vectors  $\stackrel{^}{\text{i}}$  , $\stackrel{^}{\text{j}}$ and : $\stackrel{^}{\text{i}}$  . $\stackrel{^}{\text{j}}$   = 0 $\stackrel{^}{\text{j}}$  . $\stackrel{^}{\text{k}}$   = 0 $\stackrel{^}{\text{k}}$  . $\stackrel{^}{\text{i}}$   = 0

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