Home » Blog » Find a nonzero vector satisfying a requested property

Find a nonzero vector satisfying a requested property

Let

    \[ A = (2,1,-1), \qquad B = (1,-1,2). \]

Find some vector C \neq O in \mathbb{R}^3 such that

    \[ A \cdot C = B \cdot C = 0. \]


Let C = (c_1, c_2, c_3). Then we have

    \begin{align*}  A \cdot C &= 2c_1 + c_2 - c_3 = 0\\  B \cdot C &= c_1 - c_2 + 2c_3 = 0. \end{align*}

From these equations we have

    \[ c_1 = \frac{1}{2} (c_3 - c_2) \quad \implies \quad \frac{5}{2} c_3 - \frac{3}{2}c_2 = 0 \quad \implies \quad c_2 = \frac{5}{3} c_3. \]

Letting c_3 = 3 we then have c_2 = 5 and c_1 = -1. Thus, C = (-1,5,3) is a solution.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):