Problem on cross products and the angle between vectors
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What is the angle between a nonzero vector $\bfx$ and $-\bfx$? Use that angle to show that $\bfx \times (-\bfx)=0$.
Solution
Recall that
Definition of the angle between two vectors
The angle between two vectors is the angle swept by the arc that directly connects them, provided that the vectors share the same base.
The angle between two vectors is always between zero and pi
Because $\bfx$ and $-\bfx$ point in opposite directions, the angle between them is $\pi$:
We seek a relationship between cross products and the angle between vectors.
Recall that
Cross product length and the angle between vectors
If $\theta$ is the angle between the vectors $\bfx$ and $\bfy$, then the cross product $\bfx \times \bfy$ has length $$\left| \bfx \times \bfy \right| = \left|\bfx \right| \left| \bfy \right| \sin \theta.$$
If $\bfy=-\bfx$, then $\theta = \pi.$ Because $\sin \pi = 0$, the length of $\bfx \times (- \bfx)$ is zero. Hence, $$ \bfx \times (-\bfx) = 0.$$
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