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Q.
The magnitude of the projection of the vector $a =4 i -3 j +2 k$ on the line which makes
equal angles with the coordinate axes is
EAMCETEAMCET 2011
Solution:
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 $\left.<-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right\rangle$
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 $a =4 i -3 j +2 j$ on line $v$.
$\therefore $ Projection of a along $v =\frac{ a \cdot v }{| v |}$
$=\frac{3 / \sqrt{3}}{1}=\sqrt{3}$