If the position vector of a point P with respect to origin O is î +

1.	If the position vector of a point P with respect to origin O is î + 3ĵ - 2k̂ and that of a point Q is 3î + ĵ - 2k̂, then what is the position vector of the bisector of the angle POQ? 1. î - ĵ - k̂ 2. î + ĵ - k̂ 3. î + ĵ + k̂ 4. None of the above
Answer» Correct Answer - Option 4 : None of the above Concept: A triangle ABC is said to be an isosceles triangle if triangle ABC must have two sides of equal length. Calculations: Given, the position vector of a point P with respect to origin O is î + 3ĵ - 2k̂ and that of a point Q is 3î + ĵ - 2k̂. ⇒\(\rm \bar {OP} \) = î + 3ĵ - 2k̂ and \(\rm \bar {OQ}\) = 3î + ĵ - 2k̂. ⇒ \|OP\| = \(\rm \sqrt {1+9 + 4} = \sqrt {15}\) ⇒ \|OQ\| = \(\rm \sqrt {9 + 1+ 4} = \sqrt {15}\) Here, \|OP\| = \|OQ\| \(\rm \triangle POQ\) is isoscale. The position vector of the bisector of the angle POQ = \(\rm \dfrac 1 2 (OP + OQ)\) ⇒The position vector of the bisector of the angle POQ = \(\rm \dfrac 1 2 [(î + 3ĵ - 2k̂) + (3î + ĵ - 2k̂)]\) ⇒The position vector of the bisector of the angle POQ = \(\rm \dfrac 1 2 (4î + 4ĵ - 4k̂) \) ⇒The position vector of the bisector of the angle POQ = \(\rm 2î + 2ĵ - 2k̂\)

If the position vector of a point P with respect to origin O is î + 3ĵ - 2k̂ and that of a point Q is 3î + ĵ - 2k̂, then what is the position vector of the bisector of the angle POQ? 1. î - ĵ - k̂ 2. î + ĵ - k̂ 3. î + ĵ + k̂ 4. None of the above

Answer» Correct Answer - Option 4 : None of the above

Concept:

A triangle ABC is said to be an isosceles triangle if triangle ABC must have two sides of equal length.

Calculations:

Given, the position vector of a point P with respect to origin O is î + 3ĵ - 2k̂ and that of a point Q is 3î + ĵ - 2k̂.

⇒\(\rm \bar {OP} \) = î + 3ĵ - 2k̂ and \(\rm \bar {OQ}\) = 3î + ĵ - 2k̂.

⇒ |OP| = \(\rm \sqrt {1+9 + 4} = \sqrt {15}\)

⇒ |OQ| = \(\rm \sqrt {9 + 1+ 4} = \sqrt {15}\)

Here, |OP| = |OQ|

\(\rm \triangle POQ\) is isoscale.

The position vector of the bisector of the angle POQ = \(\rm \dfrac 1 2 (OP + OQ)\)

⇒The position vector of the bisector of the angle POQ = \(\rm \dfrac 1 2 [(î + 3ĵ - 2k̂) + (3î + ĵ - 2k̂)]\)

⇒The position vector of the bisector of the angle POQ = \(\rm \dfrac 1 2 (4î + 4ĵ - 4k̂) \)

⇒The position vector of the bisector of the angle POQ = \(\rm 2î + 2ĵ - 2k̂\)