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UPSC NDA Math Model Paper 4

Comprehension:(Que No. 106 - 107)

Let a vector

\vec{r}

make angles 60°, 30° with x and y-axes respectively.

Question : 106 of 120

Marks: +1, -0

What angle does

\overset{\rightarrow}{r}

make with z-axis?

Solution:

Let l be the direction cosine along with x-axis, m be the direction cosine along the y-axis.

Therefore,

l = \cos 60^{\circ}

=\frac{1}{2}

Similarly,

m=\cos 30^{\circ}

=\frac{\sqrt{3}}{2}

We will substitute these values in (1).

l^{2}+m^{2}+n^{2}=1

\left(\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}+n^{2}=1

\frac{1}{4}+\frac{3}{4}+n^{2}=1

n^{2}=0

n=0

Therefore, the angle made with the z-axis is cos inverse of 0.

We know that

\cos 90^{\circ}=0

Hence, the angle made with the z-axis is 90°.

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