# If $A\left( { - 4,0,3} \right)$and $B\left( {14,2, - 5} \right)$, 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. $\left( {2,2,4} \right)$

B. $\left( {2,11,5} \right)$

C. $\left( { - 3, - 3, - 6} \right)$

D. $\left( {1,1,2} \right)$

Last updated date: 18th Mar 2023

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Answer

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Hint: Assume a point which will be the bisector of the given angle and then use the distance formula to obtain the equation and hence obtain the points.

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

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

Let, the angle bisector be $\left( {x,y,z} \right)$.

Therefore,

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

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

$\left( {2x + 4z + 2y} \right) = 0$

Which can also be written as,

$x + y + 2z = 0$ …..(1)

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

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

Or,

$13x + y - 7z = 0$ …..(2)

So, the points in the options either lie on (1) or (2)

Answer = (A), (C) and (D)

Note: These types of questions can have more than one answer so make sure to check all the options before moving on to the next question. We started by taking the given points and using it in the distance formula to obtain equations and then checked from the options which points satisfy the given equations.

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

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

Let, the angle bisector be $\left( {x,y,z} \right)$.

Therefore,

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

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

$\left( {2x + 4z + 2y} \right) = 0$

Which can also be written as,

$x + y + 2z = 0$ …..(1)

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

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

Or,

$13x + y - 7z = 0$ …..(2)

So, the points in the options either lie on (1) or (2)

Answer = (A), (C) and (D)

Note: These types of questions can have more than one answer so make sure to check all the options before moving on to the next question. We started by taking the given points and using it in the distance formula to obtain equations and then checked from the options which points satisfy the given equations.

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