Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $\|\mathbf{v} \text{ x } \mathbf{w}\|=\sqrt{3}$.
Should I just use something random for $\mathbf{v}$ and then solve for $\mathbf{w}$?
Thank you.
Let $v = (1,0,0)$ and $w = (a,b,0)$. $$ v\cdot w = \mid\mid v \mid\mid.\mid\mid w\mid\mid\cos \dfrac{2\pi}{3} \quad \Rightarrow \quad -2a = \sqrt{a^2 + b^2} \ \Rightarrow \ 3a^2 = b^2 \quad (1) $$ But, $$ \sqrt{3} = \mid\mid v \mid\mid.\mid\mid w\mid \mid \sin \dfrac{2\pi}{3} \quad \Rightarrow \quad \sqrt{3} = \sqrt{a^2 + b^2}.\dfrac{\sqrt{3}}{2} \quad \Rightarrow \quad a^2 + b^2 = 4 \quad (2) $$ From (1) and (2), $4a^2 = 4 \ \Rightarrow \ a=\pm 1$ and $b = \pm \sqrt{3}$. Thus, $$ v = (1,0,0) \quad \text{and} \quad w = (-1,\sqrt{3},0) $$ So that the angle between $v$ and $w$ is $2\pi/3$, we must have $a = -1$