Question

if a non zero vector c is perpendicular to the given non zero vectors a and b, then find the angle between vector c and vector a x b. ​

Answer 1

$\green{\large\underline{\bf{Solution-}}}$

By definition of Cross Product of two vectors,

$\rm :\longmapsto\:\vec{a} \times \vec{b} \: gives \: a \: vector \: perpendicular \: to \: both \: \vec{a} \: and \: \vec{b}.$

$\rm :\longmapsto\:If \: \vec{a} \times \vec{b} = \vec{c}$

$\blue{\bf :\implies\:\vec{a} \: \perp \: \vec{c} \: \: \: and \: \: \: \: \vec{b} \: \perp \: \vec{c}}$

And

Conversely,

$\blue{\rm :\longmapsto\:\bf If\:\vec{a} \: \perp \: \vec{c} \: \: \: and \: \: \: \: \vec{b} \: \perp \: \vec{c}}$

$\bf\implies \:\vec{c} \: \parallel \: \vec{a} \times \vec{b}$

$\bf\implies \:\vec{c} = \lambda \: (\vec{a} \times \vec{b})$

Let's solve the problem now!!!

Given that

$\blue{\bf :\mapsto\:\vec{a} \: \perp \: \vec{c} \: \: \: and \: \: \: \: \vec{b} \: \perp \: \vec{c}}$

$\bf\implies \:\vec{c} \: \parallel \: \vec{a} \times \vec{b}$

$\bf\implies \:angle \: between \: \vec{c} \: and \: \vec{a} \times \vec{b} \: is \: 0 \degree$

Additional Information :-

$\boxed{ \sf \:\vec{a}.\vec{b} = \vec{b}.\vec{a}}$

$\boxed{ \sf \:\vec{a} \times \vec{b} = - \vec{b} \times \vec{a}}$

$\boxed{ \sf \:\vec{a} \: . \: \vec{a} = { |\vec{a}| }^{2}}$

$\boxed{ \sf \:\vec{a} \times \vec{a} = 0}$

$\boxed{ \sf \:\vec{a}.\vec{b} = 0 \implies \: \vec{a} \: \perp \: \vec{b}}$

$\boxed{ \sf \:\vec{a} \times \vec{b} = 0 \implies \: \vec{a} \: \parallel \: \vec{b}}$