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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!

amWhy
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Ofir Attia
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1 Answers1

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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.
Alex Wertheim
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    Ok, after your instructions I found the vectors ( MN,NP,MP ) found that $MN*MP=0$ its enough? – Ofir Attia Jul 12 '13 at 20:02
  • That's right! You've found a pair of vectors with dot product $0$, so the angle between them is $\frac{\pi}{2}$; hence, the triangle is a right triangle. – Alex Wertheim Jul 12 '13 at 20:05