Vector Algebra Question 88
Question: The vector $ \overset{\to }{\mathop{c}}, $ directed along the bisectors of the angle between the vectors $ \overset{\to }{\mathop{a}},=7\hat{i}-4\hat{j}-4\hat{k}, $ $ \overset{\to }{\mathop{b}},=-2\hat{i}-\hat{j}+2\hat{k}, $ and $ |\overset{\to }{\mathop{c}},|=3\sqrt{6} $ is given by
Options:
A) $ \hat{i}-7\hat{j}+2\hat{k} $
B) $ \hat{i}+7\hat{j}-2\hat{k} $
C) $ \hat{i}+7\hat{j}+2\hat{k} $
D) $ \hat{i}+7\hat{j}+3\hat{k} $
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Answer:
Correct Answer: A
Solution:
- [a]
$ OQ=PQ=\lambda $ (say);
$ \overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{QP};\overrightarrow{c}=\lambda \hat{a}+\lambda \hat{b} $
Let $ \hat{a} $ and $ \hat{b} $ be unit Vectors along $ \overset{\to }{\mathop{a}}, $ and $ \overset{\to }{\mathop{b}}, $
Respectively, Then $ \hat{a}=\frac{1}{9}(7\hat{i}-4\hat{j}-4\hat{k}) $ and $ \hat{b}=\frac{1}{3}(-2\hat{i}-\hat{j}+2\hat{k}) $
The required vector $ \overset{\to }{\mathop{c}},=\lambda (\hat{a}+\hat{b}), $ where $ \lambda $ is a scalar $ \lambda ( \frac{1}{9}\hat{i}-\frac{7}{9}\hat{j}+\frac{2}{9}\hat{k} ) $
$ |\overset{\to }{\mathop{c}},{{|}^{2}}{{\lambda }^{2}}( \frac{1}{81}+\frac{49}{81}+\frac{4}{81} )=\frac{54}{81}{{\lambda }^{2}} $
$ \Rightarrow {{(3\sqrt{6})}^{2}}=\frac{54}{81}{{\lambda }^{2}} $
$ \Rightarrow {{\lambda }^{2}}=81\Rightarrow \lambda =\pm 9. $ Hence, $ \overset{\to }{\mathop{c}},=(\hat{i}-7\hat{j}+2\hat{k}) $
BETA
