Vector Algebra Question 185
Question: If a, b, c are non-coplanar unit vectors such that $ \mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}} $ , then the angle between a and b is
[IIT 1995]
Options:
A) $ \frac{\pi }{4} $
B) $ \frac{\pi }{2} $
C) $ \frac{3\pi }{4} $
D) $ \pi $
Show Answer
Answer:
Correct Answer: C
Solution:
- $ \mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\Rightarrow (\mathbf{a},.,\mathbf{c})\mathbf{b}-(\mathbf{a},.,\mathbf{b}),\mathbf{c}=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}} $
$ \Rightarrow [ (\mathbf{a},.,\mathbf{c})-\frac{1}{\sqrt{2}} ]\mathbf{b}-[ (\mathbf{a},.,\mathbf{b})+\frac{1}{\sqrt{2}} ],\mathbf{c}=0 $
$ \Rightarrow \mathbf{a},.,\mathbf{c}=\frac{1}{\sqrt{2}}, $ $ \mathbf{a},.,\mathbf{b}=-\frac{1}{\sqrt{2}} $
$ \Rightarrow ,|\mathbf{a}|,|\mathbf{c}|\cos \theta =\frac{1}{\sqrt{2}}, $ $ |\mathbf{a}|,|\mathbf{b}|\cos \varphi =-\frac{1}{\sqrt{2}} $
$ \Rightarrow \cos \theta =\frac{1}{\sqrt{2}}, $ $ \cos \varphi =-\frac{1}{\sqrt{2}}\Rightarrow \theta =\frac{\pi }{4}, $ $ \varphi =\frac{3\pi }{4}. $
BETA
