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, $\vec{a}\times \vec{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 0\\ 0& 1& 1\end{array}\right|$
$=\hat{i}(1-0)-\hat{j}(1-0)+\hat{k}(1-0)$
$=\hat{i}-\hat{j}+\hat{k}$
$\therefore$ 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.