Question

if v × w = ⟨2, - 2, - 3⟩ and v · w = 4, find tan(θ), where θ is the angle between v and w. tan(θ) =

Explanation:

Step1: Recall vector - related formulas

We know that $\mathbf{v}\cdot\mathbf{w}=\vert\mathbf{v}\vert\vert\mathbf{w}\vert\cos\theta$ and $\vert\mathbf{v}\times\mathbf{w}\vert=\vert\mathbf{v}\vert\vert\mathbf{w}\vert\sin\theta$. Then $\tan\theta=\frac{\vert\mathbf{v}\times\mathbf{w}\vert}{\mathbf{v}\cdot\mathbf{w}}$.

Step2: Substitute given values

Given $\mathbf{v}\times\mathbf{w}=\langle2, - 2,-3
angle$, then $\vert\mathbf{v}\times\mathbf{w}\vert=\sqrt{2^{2}+(-2)^{2}+(-3)^{2}}=\sqrt{4 + 4+9}=\sqrt{17}$. And given $\mathbf{v}\cdot\mathbf{w}=4$.

Step3: Calculate $\tan\theta$

Substitute $\vert\mathbf{v}\times\mathbf{w}\vert=\sqrt{17}$ and $\mathbf{v}\cdot\mathbf{w}=4$ into the formula $\tan\theta=\frac{\vert\mathbf{v}\times\mathbf{w}\vert}{\mathbf{v}\cdot\mathbf{w}}$. So $\tan\theta=\frac{\sqrt{17}}{4}$.

Answer:

$\frac{\sqrt{17}}{4}$