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 $p=-1$

Show Answer

Answer:

Correct Answer: 17. (9)

Solution:

  1. 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 $\quad 4 p^{2}+1=(p+1)^{2}+1 \Rightarrow 4 p^{2}=p^{2}+2 p+1$

$\Rightarrow 3 p^{2}-2 p-1=0 \Rightarrow(3 p+1)(p-1)=0$

$\Rightarrow \quad p=1,-1 / 3$



NCERT Chapter Video Solution

Dual Pane