My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
2026-03-26 09:17:56.1774516676
If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar
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Let $u,v$ be vectors in $\mathbf{R}^3$. Then, the dot product is given by $\|u\|\|v\|\cos\theta$. This is a real number. Is that a scalar?
Here's a nice exercise - prove statement $\mathrm{B}.$: $u\cdot(u\times v)=v\cdot(u\times v)=0$.