Vectors: Finding third point in a right angle triangle

Question

Given A(1,4) and B (3,-5) use the dot product to find point C so that triangle ABC is a right angle triangle.

Answer 1

Find the vector connecting between the two:

$$\vec{c}=<1-3,4+5>=<-2,9>$$

You can use the dot product between $\vec{c}$ and a unit vector $\hat{b}=<-1,0>$: $$\cos{\theta}=\frac{\vec{c}\cdot\hat{b}}{||\vec{c}||\space||\hat{b}||}$$

The length of the opposing two angles are $||\vec{c}||\sin{\theta}$ and $||\vec{c}||\cos{\theta}$.

Answer 2

Just put $(C-A).(B-A)=0$, that should solve it. Note that there are infinitely many such triangles.