A vector a has components 3 p and 1 with respect to a rectangular… - Vidyadip

MCQ

A vector $\vec a$ has components $3 p$ 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 new system, $\overrightarrow{\text { a }}$ has components $p +1$ and $\sqrt{10},$ then a value of $p$ is equal to

A
$1$
B
$-\frac{5}{4}$
C
$\frac{4}{5}$
✓
$-1$

✓

Answer

Correct option: D.

$-1$

d
$\overrightarrow{ a }_{ Old }=3 p \hat{ i }+\hat{ j }$

$\overrightarrow{ a }_{ New }=( p +1) \hat{ i }+\sqrt{10} \hat{ j }$

$\Rightarrow\left|\overrightarrow{ a }_{ Od }\right|=\left|\overrightarrow{ a }_{ New }\right|$

$\Rightarrow 9 p^{2}+1=p^{2}+2 p+1+10$

$8 p^{2}-2 p-10=0$

$4 p^{2}-p-5=0$

$(4 p-5)(p+1)=0 \rightarrow p=\frac{5}{4},-1$

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