In a triangle ABC , right angled at the vertex A , if the position vectors of A , B and C are respectively `3 hati + hatj-hatk , -hati + 3hatj + phatk` and `5 hati + q hatj - 4 hatk` , then the point (p,q) lies on a line

Options

A. parallel to y-axis
B. making an acute angle with the positive direction of x-axis
C. parallel to x-axis
D. making an obtuse angle with the positive direction of x axis

Explanation

To solve the problem step by step, we will analyze the position vectors of points A, B, and C in triangle ABC, which is right-angled at vertex A.

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
- Let the position vector of point A be \( \vec{A} = 3\hat{i} + \hat{j} - \hat{k} \).
- Let the position vector of point B be \( \vec{B} = -\hat{i} + 3\hat{j} + p\hat{k} \).
- Let the position vector of point C be \( \vec{C} = 5\hat{i} + q\hat{j} - 4\hat{k} \).

2. **Calculate the Vector \( \vec{AB} \)**:
\[ \vec{AB} = \vec{B} - \vec{A} = (-\hat{i} + 3\hat{j} + p\hat{k}) - (3\hat{i} + \hat{j} - \hat{k}) \]
\[ = (-1 - 3)\hat{i} + (3 - 1)\hat{j} + (p + 1)\hat{k} = -4\hat{i} + 2\hat{j} + (p + 1)\hat{k} \]

3. **Calculate the Vector \( \vec{AC} \)**:
\[ \vec{AC} = \vec{C} - \vec{A} = (5\hat{i} + q\hat{j} - 4\hat{k}) - (3\hat{i} + \hat{j} - \hat{k}) \]
\[ = (5 - 3)\hat{i} + (q - 1)\hat{j} + (-4 + 1)\hat{k} = 2\hat{i} + (q - 1)\hat{j} - 3\hat{k} \]

4. **Use the Right Angle Condition**:
Since triangle ABC is right-angled at A, the dot product of vectors \( \vec{AB} \) and \( \vec{AC} \) must be zero:
\[ \vec{AB} \cdot \vec{AC} = 0 \]
\[ (-4\hat{i} + 2\hat{j} + (p + 1)\hat{k}) \cdot (2\hat{i} + (q - 1)\hat{j} - 3\hat{k}) = 0 \]

5. **Calculate the Dot Product**:
\[ -4 \cdot 2 + 2 \cdot (q - 1) + (p + 1)(-3) = 0 \]
\[ -8 + 2(q - 1) - 3(p + 1) = 0 \]
\[ -8 + 2q - 2 - 3p - 3 = 0 \]
\[ 2q - 3p - 13 = 0 \]

6. **Rearranging the Equation**:
Rearranging gives us:
\[ 3p - 2q + 13 = 0 \]

7. **Identify the Line Equation**:
The equation \( 3p - 2q + 13 = 0 \) represents a line in the \( (p, q) \) coordinate system.

### Conclusion:
The point \( (p, q) \) lies on the line described by the equation \( 3p - 2q + 13 = 0 \).