1. ## Another complex number

In an Argand diagram O is the origin, P represents the number 7 – i and Q represent the number 12 + 4i.

Calculate the size of angle OPQ.

I’ve done 1st part but how can I calculate angle OPQ in this case?

2. Originally Posted by geton
In an Argand diagram O is the origin, P represents the number 7 – i and Q represent the number 12 + 4i.

Calculate the size of angle OPQ.

I’ve done 1st part but how can I calculate angle OPQ in this case?
What's the first part?

I'd call the point P(7, -1) and Q(12, 4) and use them as vectors based at the origin. Then use the dot product to get the angle.

The other way is a bit messier: put each point into the form $r~e^{i~\theta}$. The find the difference in the arguments.

-Dan

3. Originally Posted by geton
In an Argand diagram O is the origin, P represents the number 7 – i and Q represent the number 12 + 4i.

Calculate the size of angle OPQ.

I’ve done 1st part but how can I calculate angle OPQ in this case?
You can find out the modulus and arg of P and Q.
After that,you have to find the angle POQ.
Then, use the cosine formula to calculate length PQ.
Last step is use sine formula to get the angle OPQ.

4. Fro any two non-zero complex numbers z & w the angle between them can be found by: $\Theta = \arccos \left( {\frac{{{\mathop{\rm Re}\nolimits} (z){\mathop{\rm Re}\nolimits} (w) + {\mathop{\rm Im}\nolimits} (z){\mathop{\rm Im}\nolimits} (w)}}{{\left| z \right|\left| w \right|}}} \right)$