Vectors and Cross Product

Question

I have these two questions regarding the Cross Product.

1.) You are looking down at a map. A vector $u$ with $|u| = 3$ points north and a vector $v$ with $|v| = 10$ points northeast.

What is $|u \times v|$ = ?

2.) If $v \times w = \langle −1,0,5 \rangle$ and $v \cdot w = 4$, find $\tan(x)$, where $x$ is the angle between $v$ and $w$. Find the angle.

I'm confused as to how I should set up $| u \times v | = |u| |v| \sinθ$ in order to solve these two questions. Any ideas?

Answer 1

Hint:angle between v and V is $45^0$ and $$|u| = 3|v| = 10 $$ then put in and find $u\times v$

$$\lvert u\times v\rvert = \lvert u\rvert\lvert b\rvert \lvert\sin(\theta)\rvert $$ for second part use: $$\frac{|u\times v|}{|u.v|}=tan(\theta) $$

Answer 2

Don't insist on $|u||v|sin\theta$. It's not appropriate for some things, and in that case it gets in the way. For $\vec{a}=(a_1,a_2,a_3)$ Try $\vec{a}\times\vec{b}=(a_2b_3-a_3b2, a_3b_1-a_1b_3, a_1b_2-a_2b_1).$

For the second question, the dot product likewise has a trigonometric form.