Find $(\overrightarrow{a}\times\overrightarrow{b})\cdot \overrightarrow{c}$

Question

$\overrightarrow{a},\overrightarrow{b},$ and $\overrightarrow{c}$ be 3 non coplanar unit vector such that angle between every pair of them is $\frac{\pi}{3}$.

Find $(\overrightarrow{a}\times\overrightarrow{b})\cdot \overrightarrow{c}$

Approach:

$(\overrightarrow{a}\times\overrightarrow{b})\cdot \overrightarrow{c}=(|\overrightarrow{a}||\overrightarrow{b}|\sin{\theta})(\hat{n}\cdot \overrightarrow{c})$

$\implies |\overrightarrow{a}||\overrightarrow{b}| |\overrightarrow{c}|\sin{\theta} \cos(\alpha)$.

Doubt: How to find angle between unit vector $\hat{n}$ and $\overrightarrow{c}$ which is $\alpha$ .

I know other method to solve this question but i want to approach using this method only.

I check for duplicate. I didn't find any question having same doubt as me.

Answer 1

Hint: Construct a tetrahedron with vectors at a vertex of base (say $P$), being: $\mathbf {a,b,c}$. Let the base of this tetrahedron have vectors $\mathbf {a,b}$, and let the base be placed parallel to the ground. Then, we wish to find angle between $\mathbf c$ and the vertical ($\alpha$). This angle would be equal to $90°-$(Angle between base and $\mathbf c$). Now, drop a perpendicular from the top of the tetrahedron (say $O$), let it meet base at $Q$, and use elementary trigonometry for right triangle $OQP$ to find out the angle between $\mathbf c$ and base.

Answer 2

$\hat{n}$ perpendicular to the plane span by $\vec{a}$ and $\vec{b}$ angle between plane and $\vec{c}$ is as $\pi/3$, thus $\alpha = \pi/6$