# Thread: Angle in a triangle with three coordinates in space

1. ## Angle in a triangle with three coordinates in space

$\displaystyle P$, $\displaystyle Q$ and $\displaystyle R$ are three points in space with coordinates $\displaystyle (2, -1, 4)$ , $\displaystyle (3, 1, 2)$ and $\displaystyle (-1, 2, 5)$ respectively. Find angle $\displaystyle Q$ in the triangle $\displaystyle PQR$.

2. Use the dot product!

3. Im not sure how!

There are three coordinates here, but the dot product can only be used with two because the dot product is not associative?

I tried using the dot product with the vector equations of point P and R (assuming Q is in the middle?) ... I don't seem to get the right equation though. Not sure what I'm doing wrong !

4. Draw a picture, it helps.

Let $\displaystyle v$ be the vector $\displaystyle QP$ and $\displaystyle u$ the vector $\displaystyle QR$. Surely you know how to find these vectors from the points. Then $\displaystyle u\cdot v = |u| |v| \cos \theta$ where $\displaystyle \theta$ is the angle you are looking for.

Try it and post what you're doing, I'll tell you if it's right.