Question

A vector v is equally inclined to the $ x $ -axis, $ y $ -axis and $ z $ -axis respectively, its direction cosines are

• A. $ < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} > $
• B. $ < -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} > $
• C. $ < \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} > $ or $ < -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} > $
• D. None of the above

Answer 1

Answer: (C)

Let the vector $v$ make an angle $\alpha$ with each of
the three axes, then direction cosine of $v$ are
$<\cos \alpha, \cos \alpha, \cos \alpha> $
Also, $\cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha=1$
$\Rightarrow \cos ^{2} \alpha=1 / 3$
$\Rightarrow \cos \alpha=\pm \frac{1}{\sqrt{3}}$
Hence, direction cosine of $v$ are
$<\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}>$
or $<-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}> $