Vector Algebra Question 119
Question: If $ \vec{a}=\vec{i}+2\hat{j}-3\hat{k} $ and $ \vec{b}=3\hat{i}-\hat{j}+\lambda \hat{k}, $ and $ (\vec{a}+\vec{b}) $ is perpendicular to $ \vec{a}-\vec{b} $ , then what is the value of $ \lambda $ ?
Options:
A) -2 only
B) $ \pm 2 $
C) 3 only
D) $ \pm 3 $
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] As given: $ \overset{\to }{\mathop{a}},=\hat{i}+2\hat{j}-3\hat{k} $ and $ \overset{\to }{\mathop{b}},=3\hat{i}-\hat{j}+\lambda \hat{k} $ $ \overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},=\hat{i}+2\hat{j}-3\hat{k}+3\hat{i}-\hat{j}+\lambda \hat{k} $ $ =4\hat{i}+\hat{j}+(\lambda -3)\hat{k} $ and $ \overset{\to }{\mathop{a}},-\overset{\to }{\mathop{b}},=\hat{i}+2\hat{j}-3\hat{k}-3\hat{i}+\hat{j}-\lambda \hat{k} $ $ =-2\hat{i}+3\hat{j}-(3+\lambda )\hat{k} $ $ (\overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},) $ is perpendicular to $ (\overset{\to }{\mathop{a}},-\overset{\to }{\mathop{b}},) $
$ \Rightarrow (\overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},).(\overset{\to }{\mathop{a}},-\overset{\to }{\mathop{b}},)=0 $
$ \Rightarrow {4\hat{i}+\hat{j}+(\lambda -3)\hat{k}}{-2\hat{i}+3\hat{j}-(3-\lambda )\hat{k}}=0 $
$ \Rightarrow -8+3+(3^{2}-{{\lambda }^{2}})=0 $
$ \Rightarrow -4-{{\lambda }^{2}}=0 $
$ \Rightarrow \lambda =\pm ,2 $
BETA
