Given three vectors, find all vectors satisfying given relations

by RoRi

April 27, 2016

Let

$A = (2,-1,1), \qquad B = (1,2,-1), \qquad C = (1,1,-2)$

be three vectors in $\mathbb{R}^3$ . Find every vector $D$ of the form

$D = xB + yC$

where $D$ is orthogonal to $A$ and has unit length.

Since $D = xB = yC$ we have

$D = (x+y, 2x+y, -x-2y).$

Since $D$ is orthogonal to $A$ we have

$D \cdot A = 0 \quad \implies \quad 2(x+y) - (2x+y) + (-x-2y) = 0 \quad \implies \quad x = -y.$

Therefore, $D = (0,x,x)$ . Finally, since $D$ has unit length we have

$\lVert D \rVert = 1 \quad \implies \quad \sqrt{2x^2} = 1 \quad \implies \quad x =\pm \frac{1}{\sqrt{2}}.$

Therefore,

$D = \pm \frac{1}{\sqrt{2}} (0,1,1).$

Stumbling Robot