"Given $|\vec x|=2, |\vec y|=3$ and the angle between them is 120°, determine the unit vector in the opposite direction of $|\vec x – \vec y|$."
To solve this problem, would I need to use sine law or cosine law, or would I need to rewrite the vectors as Cartesian vectors, so that they are easier to calculate? I thought the answer was $-\frac 1 2 \vec x + \frac 1 3 \vec y$ at first, but that doesn't seem right, given the angle. Could you please show me how to find the solution to this problem? Thanks!
Best Answer
HINT
Denote the origin by $O = (0,0)$ and consider the vectors $x = (2,0)$ and $y = (3\cos(\theta),3\sin(\theta))$.
Based on such considerations, we are able to find the vector $v$ in the exercise as follows:
\begin{align*} v = -\frac{x - y}{\|x - y\|} = -\frac{(2 - 3\cos(\theta),-3\sin(\theta))}{\sqrt{13 - 12\cos(\theta)}} \end{align*}
Can you take it from here?