Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am working on a question from the first chapter of a Linear Algebra textbook I'm reading.

Let $A=(1,1,-1)$, $B=(-3,2,-2)$, and $C=(2,2,-4)$. Prove that $\Delta ABC$ is a right-angled triangle.

I know that the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$ must be $90°$. In other words, $\overrightarrow{AB} \cdot \overrightarrow{AC}=0$. My trouble is I cant translate these given points into vectors in order to show the necessary calculations. I'm having the same trouble with other questions. Any help would be appreciated.

share|cite|improve this question
Shift things so that the right angle is at the origin before taking dot products. – J. M. Jul 15 '12 at 18:00
up vote 3 down vote accepted

$$\vec{AB} = (-3,2,-2) - (1,1,-1) = (-4,1,-1)$$ $$\vec{BC} = (2,2,-4) - (-3,2,-2) = (5,0,-2)$$ $$\vec{CA} = (1,1,-1) - (2,2,-4) = (-1,-1,3)$$ Now look at $\vec{AB} \cdot \vec{CA}$ to conclude that the right angle is at $A$.

share|cite|improve this answer


If $U$ and $V$ are two different vectors, then the vector from $U$ to $V$ is given by $V-U$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.