Question

The number of vectors of unit length perpendicular to vectors $\mathbf{a = (1,1,0)}$ and $\mathbf{b = (0,1,1)}$ is

• A. Three
• B. One
• C. Two
• D. Infinite

Answer 1

Answer: (C)

Two

Answer 2

The vector perpendicular to $\mathbf{a}$and $\mathbf{b}$ is

\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} \right| = \mathbf{i} - \mathbf{j} + \mathbf{k}$$ Since the length of this vector is $\sqrt{3},$ the unit vector perpendicular to $\mathbf{a}$ and $\mathbf{b}$ is $\pm \frac{\mathbf{a} \times \mathbf{b}}{|\mathbf{a} \times \mathbf{b}|} = \pm \frac{1}{\sqrt{3}}(\mathbf{i} - \mathbf{j} + \mathbf{k})$ Hence there are two such vectors.