Proving a triangle is a right triangle given vertices, using vector dot product

Question

I want to to show that this triangle is a right triangle.

I know that the dot of the vectors need to be $0.$ I tried to dot between them but I don't get zero.

Claim: Triangle $\bigtriangleup MNP,\;\;\,M(1,-2,3),\;\;N(0,0,4),\;\; P(4,2,-2)\;$ is a right triangle.

What did I do wrong? I would like to get some hints how to do it.
Thanks!

Answer 1

Hints:

• What you have listed there are three ${\bf points}$. You need three ${\bf vectors}$. How can you get the three vectors associated with each side of the triangle from these three points? (sub-hint: how do you get a vector between any two points?)
• Furthermore, you need to check if the dot product is zero between any pair of sides. Since the dot product is symmetric in $\mathbb{R}^{3}$, you only have to check this for three pairs of vectors. Using this, you should be able to demonstrate that there is indeed a pair of vectors whose dot product is $0$, therefore showing that the triangle is right.