Question

The number of vectors of unit length perpendicular to the two vectors $ a=(1,1,0) $ and $ b=(0,1,1) $ is:

• A. one
• B. two
• C. three
• D. infinite

Answer 1

Answer: (B)

We have $\vec{a}=\hat{i}+\hat{j}, \hat{b}=\hat{j}+\hat{k}$
Now, $ \vec{a}\times \vec{b}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{matrix} \right| $
$=\hat{i}(1-0)-\hat{j}(1-0)+\hat{k}(1-0)$
$=\hat{i}-\hat{j}+\hat{k}$
$\therefore $ Unit vectors $=\pm \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$
$=\pm \frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{1+1+1}}=\pm \frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$
So, there are two unit length perpendicular to the two vectors.