MHT CET · Maths · Three Dimensional Geometry
A vector \(\bar{a}\) has components \(2 p\) and 1 with respect to a rectangular Cartesian system. This system is rotated though a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, à has components \(p+1\) and 1 , then
- A \(p=0\)
- B \(\mathrm{p}=-1\) or \(\mathrm{p}=\frac{1}{3}\)
- C \(\mathrm{p}=1\) or \(\mathrm{p}=-\frac{1}{3}\)
- D \(\mathrm{p}=1\) or \(\mathrm{p}=-1\)
Answer & Solution
Correct Answer
(C) \(\mathrm{p}=1\) or \(\mathrm{p}=-\frac{1}{3}\)
Step-by-step Solution
Detailed explanation
Magnitude of \(\vec{a}\) before rotation = Magnitude of \(\vec{a}\) after rotation
\(\begin{aligned} & \Rightarrow(2 p)^2+1^2=(p+1)^2+1^2 \\ & \Rightarrow 4 p^2+1=p^2+2 p+1+1 \\ & \Rightarrow 3 p^2-2 p-1=0 \\ & \Rightarrow p=1 \text { or } p=\frac{-1}{3}\end{aligned}\)
\(\begin{aligned} & \Rightarrow(2 p)^2+1^2=(p+1)^2+1^2 \\ & \Rightarrow 4 p^2+1=p^2+2 p+1+1 \\ & \Rightarrow 3 p^2-2 p-1=0 \\ & \Rightarrow p=1 \text { or } p=\frac{-1}{3}\end{aligned}\)
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