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 :

  • 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

 

Answers (1)

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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