Vectors 1 Question 16
17. A vector $\overrightarrow{\mathbf{a}}$ has components $2 p$ and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, $\overrightarrow{\mathbf{a}}$ has components $p+1$ and 1 , then
(1986, 2M)
(a) $p=0$
(b) $p=1$ or $p=-\frac{1}{3}$
(c) $p=-1$ or $p=\frac{1}{3}$
(d) $p=1$ or $p=-1$
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Answer:
Correct Answer: 17. (9)
Solution:
- Here, $\overrightarrow{\mathbf{a}}=(2 p) \hat{\mathbf{i}}+\hat{\mathbf{j}}$, when a system is rotated, the new component of $\overrightarrow{\mathbf{a}}$ are $(p+1)$ and 1 .
i.e. $\quad \overrightarrow{\mathbf{b}}=(p+1) \hat{\mathbf{i}}+\hat{\mathbf{j}} \Rightarrow|\overrightarrow{\mathbf{a}}|^{2}=|\overrightarrow{\mathbf{b}}|^{2}$
or $\quad 4 p^{2}+1=(p+1)^{2}+1 \Rightarrow 4 p^{2}=p^{2}+2 p+1$
$\Rightarrow 3 p^{2}-2 p-1=0 \Rightarrow(3 p+1)(p-1)=0$
$\Rightarrow \quad p=1,-1 / 3$
