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STD 12 - 10. vector algebra — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 10. vector algebra1 Mark
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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