Question

If $A(-4,0,3)$ and $B(14,2,-5)$, then which one of the following points lie on the bisector of the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ (O is the origin of reference)?

• A. $(2,2,4)$
• B. $(2,11,5)$
• C. $(-3,-3,-6)$
• D. $(1,1,2)$

Answer 1

Answer: (A), (C), (D)

The correct options are
A $(2,2,4)$
C $(-3,-3,-6)$
D $(1,1,2)$

Given $\overrightarrow{A}(-4,0,3)$

$\overrightarrow{B}(14,2,-5)$

On the angle bisector be $⟨x,y,z⟩$

$\cos {\theta }_A=\left|\frac{-4x+3z}{\sqrt{25}}\right|=\cos {\theta }_B=\left|\frac{14x+2y-5z}{\sqrt{225}}\right|$

or $(-4x+3z)3=-14x-2y+5z$

$2x+4z+2y=0$

$x+y+2z=0-\left(1\right)$

$\left(3\right)(-4x+3z)=14x+2y-5z$

$26x+2y-14z=0$

or $13x+y-7z=0-\left(2\right)$

So the points in the options either lie on $\left(1\right)$ or $\left(2\right)$

$\left(A\right)(2,2,4)$ lies on $\left(2\right)$

$\left(C\right)(-3,-3,-6)$ lies on $\left(2\right)$

$\left(D\right)(1,1,2)$ lies on $\left(2\right)$