[Math] Cross product of two vectors, given magnitudes and angle

Question

Problem

Two vectors $\mathopen|{\overrightarrow{a}|=5.39} \ and \ \mathopen|{\overrightarrow{b}|=4.65} $ intersect and make a 120° angle. Find $\mathopen|{\overrightarrow{a}}\times \mathopen{\overrightarrow{b}|}$

Now I tried to solve this problem for too much time and since I have the solution I've seen that the result is $-12.5$ and in particular $-12.5 =\mathopen|{\overrightarrow{a}|}\cdot \mathopen|{\overrightarrow{b}| \cdot \cos120}$

Could please somebody show me how to properly solve this problem? Thanks in advice

With notation $\mathopen|{\overrightarrow{a}|}=5.39$ I mean the magnitude and not the coordinates

Answer 1

Hint: Recall definition of cross product of $2$ vectors in $\mathbb{R}^n$ is:

$|\overrightarrow{a}\times \overrightarrow{b}| = |\overrightarrow{a}|\cdot |\overrightarrow{b}|\cdot \sin (\overrightarrow{a},\overrightarrow{b})$. Can you take it from here.