Browse Questions

# Find the value of $x$ for which $x(\hat i + \hat j +\hat k)$ is a unit vector.

$\begin{array}{1 1} (A) x = \pm \large\frac{1}{\sqrt 3} \\ (B) x = \pm \large\frac{1}{3} \\ (C) x = \pm 1 \\ (D) x = \pm 3\end{array}$

Toolbox:
• If $\overrightarrow a=a_1\hat i+a_2\hat j+a_3\hat k$
• $|\overrightarrow a|=\sqrt{a_1^2+a_2^2+a_3^2}$
Step 1:
Let $\overrightarrow a=x(\hat i+\hat j+\hat k)$
$\qquad\;\;=x\hat i+x\hat j+x\hat k$
The magnitude of $\overrightarrow a$ is $|\overrightarrow a|=\sqrt{x^2+x^2+x^2}$
$\qquad\qquad\qquad\qquad\quad\;\;=\sqrt{3x^2}$
$\qquad\qquad\qquad\qquad\quad\;\;=x\sqrt{3}$
Step 2:
But it is given $\overrightarrow a$ is a unit vector.
Hence $|\overrightarrow a|=1$
$\Rightarrow x\sqrt 3=\pm 1$
Therefore $x=\pm\large\frac{1}{\sqrt 3}$