Q. 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 :
Solution:
$\overline{AB}=-4\hat{i}+2\hat{j}+\left(p+1\right)\hat{k}$
$\overline{AB}=2\hat{i}+\left(a_{n}-1\right)\hat{J}-3\hat{k}$
$\overline{AB} \bot\overline{AC}$
$\Rightarrow \overline{AB}. \overline{AC}=0$
$-8+2\left(a_{n}-1\right)-3\left(p+1\right)=0$
$3p-2q+13=0$
$\left(p, q\right)$lies on $3x-2y+13=0$
slope$=\frac{3}{2}$
$\therefore $ Acute angle with x - axis
