QJEEM18Ma877c: JEE Main 2018 Physics - Class 11 JEEM18Ma877c Solution: If the position vectors of the vertices A, B and C of a ABC are …

Question

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If the position vectors of the vertices A, B and C of a

\Delta

ABC are respectively

4\widehat i + 7\widehat j + 8\widehat k,

2\widehat i + 3\widehat j + 4\widehat k,

and

2\widehat i + 5\widehat j + 7\widehat k,

then the position vectors of the point, where the bisector of

\angle

A meets BC is :

Accepted Answer

Suppose angular bisector of A meets BC at D(x, , z)

Using angular bisector theorem,

{{AB} \over {AC}}

=

{{BD} \over {DC}}

{{BD} \over {DC}}

=

{{\sqrt {{{\left( {4 - 2} \right)}^2} + {{\left( {7 - 3} \right)}^2} + {{\left( {8 - 4} \right)}^2}} } \over {\sqrt {{{\left( {4 - 2} \right)}^2} + {{\left( {7 - 5} \right)}^2} + {{\left( {8 - 7} \right)}^2}} }}

=

{{\sqrt {{2^2} + {4^2} + {4^2}} } \over {\sqrt {{2^2} + {2^2} + {1^2}} }}

=

{6 \over 3}

= 2

So, D(x, y, z)

\equiv

\left( {{{\left( 2 \right)\left( 2 \right) + \left( 1 \right)\left( 2 \right)} \over {2 + 1}},{{\left( 2 \right)\left( 5 \right) + \left( 1 \right)\left( 3 \right)} \over {2 + 1}}} \right.

,

\left. {{{\left( 2 \right)\left( 7 \right) + \left( 1 \right)\left( 4 \right)} \over {2 + 1}}} \right)

D(x, y, z)

\equiv

\left( {{6 \over 3},{{13} \over 3},{{18} \over 3}} \right)

Therefore, position vector of point p =

{1 \over 3}

(6i + 13j + 18k)

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