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Question

Mathematics Question on introduction to three dimensional geometry

If the direction consines of a vector of magnitude 3 are 23,−a3,23,a>0\frac{2}{3 }, \frac{-a}{3},\frac{2}{3},a>0 then the vector is

A

2i^+j^+2k^2\hat{i}+\hat{j}+2\hat{k}

B

2i^−j^+2k^2\hat{i}-\hat{j}+2\hat{k}

C

i^−2j^+2k^\hat{i}-2\hat{j}+2\hat{k}

D

i^+2j^−2k^\hat{i}+2\hat{j}-2\hat{k}

Answer

2i^−j^+2k^2\hat{i}-\hat{j}+2\hat{k}

Explanation

Solution

Required vector = 3(l i^+m j^+n k^)3 \left( l \, \hat{i} + m \, \hat{j} + n \, \hat{k}\right) Where l2+m2+n2l^2 + m^2 +n^2 = 1 ⇒\Rightarrow 49+a29+49=1\frac{4 }{9} + \frac{a^2}{9} + \frac{4}{9} =1 ⇒\Rightarrow a29+89=1\frac{a^2}{9} + \frac{8}{9} =1 ⇒\Rightarrow a2=1 ⇒ a=1a^2 = 1 \,\Rightarrow \, a = 1 Required vector = 3 (23 i^−13 j^+23 k^) \left( \frac{2}{3} \, \hat{i} - \frac{1}{3}\, \hat{j} + \frac{2}{3} \, \hat{k}\right) = 2 i^−j^+2 k^2\,\hat{i} -\hat{j} +2 \, \hat{k}