Question

The magnitude of the projection of the vector $a=4i-3j+2k$ on the line which makes equal angles with the coordinate axes is

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (B)

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, ${\cos }^2\alpha +{\cos }^2\alpha +{\cos }^2\alpha =1$
Hence, direction cosine of $v$ are
$<\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}>$ or $<-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}⟩$
So, the required line which makes equal angle with the coordinate axes is
$v=\pm \frac{1}{\sqrt{3}}i\pm \frac{1}{\sqrt{3}}j\pm \frac{1}{\sqrt{3}}k$
Now, the magnitude of the projection of the vector $a=4i-3j+2j$ on line $v$.
$\therefore$ Projection of a along $v=\frac{a\cdot v}{|v|}$
$=\frac{3/\sqrt{3}}{1}=\sqrt{3}$