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Find all orthogonal pairs of vectors amongst a given set of vectors

by RoRi

Consider the vectors in \mathbb{R}^3,

    \[ A = (4,1,-3), \quad B = (1,2,2), \quad C = (1,2,-2), \quad D = (2,1,2), \quad E = (2,-2,-1). \]

Find all pairs which are orthogonal.

We know that two vectors are orthogonal if and only if their dot product is 0. So, we compute the dot products of all of the pairs,

    \begin{align*} A \cdot B &= 4 + 2 - 6 = 0 \\ A \cdot C &= 4 + 2 + 6 = 12 \\ A \cdot D &= 8 + 1 - 6 = 3 \\ A \cdot E &= 8 - 2 + 6 = 12 \\ B \cdot C &= 1 + 4 - 4 = 1 \\ B \cdot D &= 2 + 2 + 4 = 8 \\ B \cdot E &= 2 -4 - 2 = -4 \\ C \cdot D &= 2 +2 - 4 = 0 \\ C \cdot E &= 2 - 4 +2 = 0 \\ D \cdot E &= 4 - 2 - 2 = 0. \end{align*}

Therefore, the pairs (A,B), \ (C,D), \ (C,E), \ (D,E) are the only ones which are orthogonal.

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