Vector Algebra Question 163
Question: A vector a has components 2p and 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If a has components p+1 and 1 with respect to the new system, then
[IIT 1984]
Options:
A) $ p=0 $
B) $ p=1 $ or $ -\frac{1}{3} $
C) $ p=-1 $ or $ \frac{1}{3} $
D) $ p=1 $ or $ -1 $
Show Answer
Answer:
Correct Answer: B
Solution:
- If $ x,y $ are the original components; $ X,Y $ the new components and $ \alpha $ is the angle of rotation, then $ x=X\cos \alpha -Y\sin \alpha $ and $ y=X\sin \alpha +Y\cos \alpha $
$ \therefore ,2p=(p+1)\cos \alpha -\sin \alpha $ and $ 1=(p+1)\sin \alpha +\cos \alpha $
Squaring and adding, we get $ 4p^{2}+1={{(p+1)}^{2}}+1 $
$ \Rightarrow p+1=\pm 2p\Rightarrow p=1 $ or $ -\frac{1}{3}. $
BETA
