Question

A vector which makes equal angle with the vectors $\frac{1}{3}(i-2j+2k),\frac{1}{5}(-4i-3k)$ and $j$ is

• A. $5i+j+5k$
• B. $-6i+j+5k$
• C. $5i-j-5k$
• D. $5i+j-5k$

Answer 1

Answer: (C)

$5i-j-5k$

Let the required vector be $a=xi+yj+zk$. It makes equal angles with the vectors
$b=\frac{1}{3}(i-2j+2k)$
$c=\frac{1}{5}(-4i-3k),d=j$
Therefore, $a\cdot b=a\cdot c=a\cdot d$ ....[$\because b,c,d$ are unit vectors]
If $a\cdot b=a\cdot d$, then
$\frac{1}{3}(x-2y+2z)=y$
If $a\cdot c=a\cdot d$, then
$\frac{1}{5}(-4x-3z)=y$
$\Rightarrow x-5y+2z=0$
and $4x+5y+3z=0$
Solving these equations, we get
$x=-z$ and $x=-5y$
$\therefore \frac{x}{-5}=\frac{y}{1}=\frac{z}{5}$
$\Rightarrow \frac{x}{5}=\frac{y}{-1}=\frac{z}{-5}$
$\Rightarrow a=-5i+j+5k$ or $a=5i-j-5k$