A unit vector a makes an angle 4 with z- axis. If a + i + j is a … - Vidyadip

MCQ

A unit vector $a$ makes an angle $\frac{\pi }{4}$ with $z-$ axis. If $a + i + j$ is a unit vector, then $a$ is equal to

A
$\frac{i}{2} + \frac{j}{2} + \frac{k}{{\sqrt 2 }}$
B
$\frac{i}{2} + \frac{j}{2} - \frac{k}{{\sqrt 2 }}$
✓
$ - \frac{i}{2} - \frac{j}{2} + \frac{k}{{\sqrt 2 }}$
D
None of these

✓

Answer

Correct option: C.

$ - \frac{i}{2} - \frac{j}{2} + \frac{k}{{\sqrt 2 }}$

c
(c) Let $a = li + mj + nk,$ where ${l^2} + {m^2} + {n^2} = 1.$

a makes an angle $\frac{\pi }{4}$ with $z - $axis.

$\therefore \,\,n = \frac{1}{{\sqrt 2 }},$ ${l^2} + {m^2} = \frac{1}{2}$ …..$(i)$

$\therefore \,\,a = l\,i + m\,j + \frac{k}{{\sqrt 2 }}$

$a + i + j = (l + 1)i + (m + 1)j + \frac{k}{{\sqrt 2 }}$

Its magnitude is $1,$ hence ${(l + 1)^2} + {(m + 1)^2} = \frac{1}{2}$ .....$(ii)$

From $(i)$ and $(ii),$ $2lm = \frac{1}{2} \Rightarrow l = m = - \frac{1}{2}$

Hence $a = - \frac{i}{2} - \frac{j}{2} + \frac{k}{{\sqrt 2 }}$.

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