Question

A vector $a=(x,y,z)$ of length $2\sqrt{3}$ which makes equal angles with the vectors $b=(y,-2z,3x)$ and $c=(2z,3x,-y)$ and is perpendicular to $d=(1,-1,2)$ and makes an obtuse angle with y-axis is

• A. $(-2,2,2)$
• B. $(1,1,\sqrt{10})$
• C. $(2,-2,-2)$
• D. none of these

Answer 1

Answer: (C)

$(2,-2,-2)$

Sice $a$ is perpendicular to $d$, So $x-y+2z=0$ ...(1)
Moreover $\left|b\right|=\left|c\right|$ so $a.b=a.c$ as $a$ makes equal angles with $b$ and $c$.
Thus $xy-2yz+3xz=2xz+3xy-yz$
$\Rightarrow xz-2xy-yz=0$ ...(2)
Also $x^2+y^2+z^2=12$ ...(3)
and $y<0$
Substituting the value of $y$ from (1) in (2), we get
$x^2+2xz+z^2=0$ so $x=-z$ and $y=z$
Again substituting these value in (3) we get

$z^2=4\Rightarrow z=\pm 2$ but $y<0$ and $y=z$
So $z=-2=y$ and $x=2$