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

Question

In a triangle $A B C$, 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\

Professor PC Thomas Classes · Answer

(d) : We have,\ $\overrightarrow{A B}=-4 \hat{i}+2 \hat{j}+(p+1) \hat{k}$\ $\overrightarrow{A C}=2 \hat{i}+(q-1) \hat{j}-3 \hat{k}$\ $A B C$ is a right angled triangle, right angle at A.\ $	herefore \quad \overrightarrow{A B} \perp \overrightarrow{A C} \Rightarrow \overrightarrow{A B} \cdot \overrightarrow{A C}=0$\ $\Rightarrow \quad-8+2(q-1)-3(p+1)=0$\ $\Rightarrow \quad 3 p-2 q+13=0$\ $	herefore \quad(p, q)$ lies on the line $3 x-2 y+13=0$\ Now, slope of line $=\frac{3}{2}$\ $	herefore \quad$ The point $(p, q)$ lies on a line making acute angle with the positive direction of $x$-axis.\