Exercise 1.2.31

Question

Pick any numbers that add to $x+y+z=0$. Find the angle between your vector $v=(x, y, z)$ and the vector $\boldsymbol{w}=(z, x, y)$. Challenge question: Explain why $v \cdot w /\|v\|\|w\|$ is always $-\frac{1}{2}$

Prakash Anand · Accepted Answer

For a specific example, pick $\boldsymbol{v}=(1,2,-3)$ and then $\boldsymbol{w}=(-3,1,2)$. In this example $\cos 	heta=\boldsymbol{v} \cdot \boldsymbol{w} /\|\boldsymbol{v}\|\|\boldsymbol{w}\|=-7 / \sqrt{14} \sqrt{14}=-1 / 2$ and $	heta=120^{\circ} .$ This always happens when $x+y+z=0$

$$
\boldsymbol{v} \cdot \boldsymbol{w}=x z+x y+y z=\frac{1}{2}(x+y+z)^{2}-\frac{1}{2}\left(x^{2}+y^{2}+z^{2}ight)
$$

$$
	ext { This is the same as } \boldsymbol{v} \cdot \boldsymbol{w}=0-\frac{1}{2}\|\boldsymbol{v}\|\|\boldsymbol{w}\| . 	ext { Then } \cos 	heta=\frac{1}{2}
$$

Exercise 1.2.31

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Exercise 1.2.31

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