If `3i-2j+5k and-2i+pj-qk` are collinear vectors, then

Options

A. `p=4/3,q=(-10)/(3)`
B. `p=10/3,q=4/3`
C. `p=(-4)/(3),q=(10)/(3)`
D. `p=4/3,q=10/3`

Answer

To determine the values of \( p \) and \( q \) such that the vectors \( \mathbf{a} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} \) and \( \mathbf{b} = -2\mathbf{i} + p\mathbf{j} - q\mathbf{k} \) are collinear, we can use the property of collinearity of vectors.

### Step-by-step Solution:

1. **Understanding Collinearity**:
Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are collinear if there exists a scalar \( k \) such that:
\[ \mathbf{b} = k \mathbf{a} \]
This implies that the ratios of their corresponding components must be equal:
\[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} \]

2. **Identifying Components**:
For the vectors given:
- \( \mathbf{a} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} \) has components \( a_1 = 3, a_2 = -2, a_3 = 5 \).
- \( \mathbf{b} = -2\mathbf{i} + p\mathbf{j} - q\mathbf{k} \) has components \( b_1 = -2, b_2 = p, b_3 = -q \).

3. **Setting Up the Ratios**:
From the collinearity condition, we can set up the following equations:
\[ \frac{3}{-2} = \frac{-2}{p} = \frac{5}{-q} \]

4. **Finding \( p \)**:
From the first part of the ratio:
\[ \frac{3}{-2} = \frac{-2}{p} \]
Cross-multiplying gives:
\[ 3p = 4 \implies p = \frac{4}{3} \]

5. **Finding \( q \)**:
Now, using the second part of the ratio:
\[ \frac{3}{-2} = \frac{5}{-q} \]
Cross-multiplying gives:
\[ 3(-q) = 5(-2) \implies -3q = -10 \implies q = \frac{10}{3} \]

6. **Final Values**:
Thus, the values of \( p \) and \( q \) are:
\[ p = \frac{4}{3}, \quad q = \frac{10}{3} \]

### Conclusion:
The final answer is:
- \( p = \frac{4}{3} \)
- \( q = \frac{10}{3} \)