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If 4î+7ĵ+8k̂,...

Question

If $4^i + 7^j + 8^k, 2^i + 3^j + 4^k$ and $2^i + 5^j + 7^k$ are the position vectors of the vertices $A, B$ and $C$ respectively of triangle $A B C$ . The position vector of the point where the bisector of $∠ A$ meets $B C$

A

\frac{1}{2} (4^i + 8^j + 11^k)

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B

\frac{1}{3} (6^i + 11^j + 15^k)

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C

\frac{1}{3} (6^i + 13^j + 18^k)

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D

\frac{1}{4} (8^i + 14^j + 9^k)

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Solution

The correct option is C $\frac{1}{3} (6^i + 13^j + 18^k)$
Given: $\to A = 4^i + 7^j + 8^k, \to B = 2^i + 3^j + 4^k$ and $\to C = 2^i + 5^j + 7^k$
Suppose the bisector of angle $A$ meets $- - \to B C$ at $D$ .
Then $- - \to A D$ divides $- - \to B C$ in the ratio $- - \to A B : - - \to A C$
$∴ P . V . o f D = \frac{| - - \to A C | (2^i + 3^j + 4^k) + | - - \to A B | (2^i + 5^j + 7^k)}{| - - \to A B | + | - - \to A C |}$
$- - \to A B = \to B - \to A = - 2^i - 4^j - 4^k$
$- - \to A C = \to C - \to A = - 2^i - 2^j -^k$
$| - - \to A B | = \sqrt{4 + 16 + 16} = 6$
$| - - \to A C | = \sqrt{4 + 4 + 1} = 3$
$∴ P . V . o f D = \frac{3 (2^i + 3^j + 4^k) + 6 (2^i + 5^j + 7^k)}{6 + 3}$
$∴ P . V . o f D = \frac{1}{3} (6^i + 13^j + 18^k)$

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