I have two vectors $u$, and $v$, I know that $\mid u \mid$ = 3 and $\mid v \mid$ = 5, and that $u\cdot v = -12$. I need to calculate the length of the vector $(3u+2v) \times (3v-u)$.
Because I know the dot product of the vectors I know what cosine of the angle between them is $\cos \theta = -0.8$, and also $\sin \theta = 0.6$ Using this I started calculating the components of the vectors, but got nowhere. Am I missing some sort of fast, clever way of doing this?
$\endgroup$ 22 Answers
$\begingroup$Hint 1: $u\times u=v\times v=0$
Hint 2: $|u\times v|^2=|u|^2|v|^2-(u\cdot v)^2$
$\endgroup$ $\begingroup$The magnitude of the cross product of $u$ and $v$ is equal to the area of the parallelogram determined by $u$ and $v$, $$\|u \times v\| = \|u\| \cdot \|v\| \cdot \sin \theta$$
$\endgroup$
