JEE Main 2024 (Online) 1st February Evening ShiftQuestion 1
Consider a $\triangle ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\overrightarrow{AD}$ on the vector $\overrightarrow{AC}$ is :
Vector Algebra
Solution
$AB=\sqrt{9+25+4}=\sqrt{38}$, $AC=\sqrt{4+9+25}=\sqrt{38}$. The bisector meets $BC$ at its midpoint $D=(\tfrac12,7,\tfrac72)$. $\vec{AD}=(-\tfrac12,4,\tfrac32)$, $\vec{AC}=(2,3,5)$. $\vec{AD}\cdot\vec{AC}=-1+12+\tfrac{15}2=\tfrac{37}2$. Projection length $=\dfrac{37/2}{\sqrt{38}}=\dfrac{37}{2\sqrt{38}}$. Option (B).