In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are…

Question

In a triangle $ABC$, right angled at the vertex $A$, if the position vectors of $A$, $B$ and $C$ are respectively $3\hat i+\hat j-\hat k$, $-\hat i+3\hat j+p\hat k$ and $5\hat i+q\hat j-4\hat k$, then the point $(p,q)$ lies on a line:

Accepted Answer

$\overrightarrow{AB}=-4\hat i+2\hat j+(p+1)\hat k$, $\overrightarrow{AC}=2\hat i+(q-1)\hat j-3\hat k$. Since $AB\perp AC$:

$$\overrightarrow{AB}\cdot\overrightarrow{AC}=0 \implies -8+2(q-1)-3(p+1)=0 \implies 3p-2q+13=0$$

So $(p,q)$ lies on $3x-2y+13=0$, slope $=\dfrac32>0$, which makes an acute angle with the positive $x$-axis.

Solutions · 1