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Latus Rectum

The position ...

Question

The position vectors of the points A and B w.r.t. an origin are $a = i + 3 j - 2 k$ and
$b = 3 i + j - 2 k$ respectively. Determine the vectgor $\to O P$ which bisects the angle AOB, where P is a point on AB.

A

2 (i - j + k)

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B

2 (i - j - k)

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C

2 (- i + j + k)

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D

2 (i + j - k)

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Solution

The correct option is D $2 (i + j - k)$
$∣ ∣ \to O A ∣ ∣ = ∣ ∣ \to O B ∣ ∣ = \sqrt{14}$
$△ A O B$ is isosceles.
Hence the bisector of angle $A O B$ will bisect the base $A B .$
Hence $P$ is mid-point $(2, 2, - 2)$ of $A B$
$∴ \to O P = 2 (i + j - k)$

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