3D Geometry
Angle Bisector Length in 3D Triangle
nta_pyq_2024_jan
Grade 12
Question:
The position vectors of the vertices $A$, $B$ and $C$ of a triangle are $2\hat{i}-3\hat{j}+3\hat{k}$, $2\hat{i}+2\hat{j}+3\hat{k}$ and $-\hat{i}+\hat{j}+3\hat{k}$ respectively. Let $l$ denote the length of the angle bisector $AD$ of $\angle BAC$ where $D$ is on the line segment $BC$, then $2l^2$ equals:
Step-by-Step Solution
Key Concept: $AB=5$, $AC=\sqrt{(−3)^2+4^2}=5$. Isosceles! $D$ is midpoint of $BC$. $D=((2-1)/2,(2+1)/2,(3+3)/2)=(1/2,3/2,3)$. $AD=|D-A|=|(-3/2,9/2,0)|=\sqrt{9/4+81/4}=\sqrt{90/4}=3\sqrt{10}/2$. $2l^2=2\times90/4=45$.
$2l^2=45$.
Correct Answer: 4