Question

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

• A. 1
• B. 2
• C. 3
• D. $\infty$

Answer 1

The correct option is B 2

Let the vector be $a\hat{i}+b\hat{j}+c\hat{k}$

a) $a\left(1\right)+b\left(1\right)+c\left(0\right)=0$

$\Rightarrow a+b=0$.......(1)

b) $a\left(0\right)+b\left(1\right)+c\left(1\right)=0$

$\Rightarrow b+c=0$........(2)

c) $\sqrt{a^2+b^2+c^2}=1$..........(3)

$\Rightarrow a^2+(-a{)}^2+(a{)}^2=1^2$

$3a^2=1$

$a=\pm \frac{1}{\sqrt{3}}$

$\therefore$ there are two vectors

$\overrightarrow{v_1}=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$

$\overrightarrow{v_2}=\frac{\widehat{-i}+\hat{j}-\hat{k}}{\sqrt{3}}$