# In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively  then the point (p, q) lies on a line : Option 1) parallel to x-axis. Option 2) parallel to y-axis. Option 3) making an acute angle with the positive direction of x-axis. Option 4) making an obtuse angle with the positive direction of x-axis

As we learnt in

Scalar Product of two vectors -

$\vec{a}.\vec{b}> 0 \:an\: acute\: angle$

$\vec{a}.\vec{b}< 0 \:an\: obtuse\: angle$

$\vec{a}.\vec{b}= 0 \:a\:right\: angle$

- wherein

$\Theta$  is the angle between the vectors $\vec{a}\:and\:\vec{b}$

$\overrightarrow{AB}= 4\hat{i}-2\hat{j}-\left ( p+1 \right )\hat{k}$

$\overrightarrow{AC}= 2\hat{i}+\left ( q-1 \right )\hat{j}-3\hat{k}$

$\overrightarrow{AB}.\overrightarrow{AC}=0$

$8-2q+2+3p+3=0$

$3p-2q+13=0$

Replace (p,q) with (x,y)

3x-2y+13=0

$Slope=\frac{3}{2}$

Acute angle with +x-axis

Option 1)

parallel to x-axis.

This option is incorrect.

Option 2)

parallel to y-axis.

This option is incorrect.

Option 3)

making an acute angle with the positive direction of x-axis.

This option is correct.

Option 4)

making an obtuse angle with the positive direction of x-axis

This option is incorrect.

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