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 $Î\pm$ with each of
the three axes, then direction cosine of $v$ are
$<\cos Î\pm ,\cos Î\pm ,\cos Î\pm >$
Also, ${\cos }^2Î\pm +{\cos }^2Î\pm +{\cos }^2Î\pm =1$
$⇒{\cos }^2Î\pm =1/3$
$⇒\cos Î\pm =Â\pm \frac{1}{\sqrt{3}}$
Hence, direction cosine of $v$ are
$<\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}>$
or $<−\frac{1}{\sqrt{3}},−\frac{1}{\sqrt{3}},−\frac{1}{\sqrt{3}}>$