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Given two vectors, find a point such that the three vectors form a right triangle

Let A = (2,-1,1) and B= (3,-4,-4) be vectors in \mathbb{R}^3. Find a point C \in \mathbb{R}^3 such that A,B,C are the vertices of a right triangle.


We choose C = (c_1, c_2, c_3) to be a point such that A-C is orthogonal to A-B (so we have a right angle). So,

    \begin{align*}  && (A-B) \cdot (A-C) &= 0 \\  \implies && (-1,3,-5) \cdot (2 - c_1, -1 - c_2, 1 - c_3) &= 0 \\  \implies && c_1 - 2 - 3 - 3c_2 -5 + 5c_3 &= 0 \\  \implies && -10 + c_1 - 3c_2 + 5c_3 &= 0 \\  \implies && c_1 &= 3c_2 - 5c_3 + 10. \end{align*}

The choices of c_2 and c_3 are arbitrary, so let’s take c_2 = c_3 = 1, then we have c_1 = 8. Therefore C = (8,1,1) is a solution.

One comment

  1. Francisco says:

    the vector formed by (A-B)=(-1,3,5), right? Why in the resolution the results is “(-1,3,-5)”, correct me if I’m wrong please, I’m VEEEEEEERY confused.

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