Given two vectors `a = 2i -3j+6k, b=2i+2j-k` and `p = ("the projection of b on a ")/("the projection of a on b")`, then the value of p is

Options

A. `3//7`
B. `7//3`
C. 3
D. 7

Answer

To solve the problem, we need to find the value of \( p \), which is defined as the ratio of the projection of vector \( b \) on vector \( a \) to the projection of vector \( a \) on vector \( b \).

### Step-by-Step Solution

1. **Define the Vectors**:
Given:
\[ \mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k} \]
\[ \mathbf{b} = 2\mathbf{i} + 2\mathbf{j} - \mathbf{k} \]

2. **Calculate the Magnitude of \( \mathbf{a} \)**:
\[ |\mathbf{a}| = \sqrt{(2)^2 + (-3)^2 + (6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

3. **Calculate the Magnitude of \( \mathbf{b} \)**:
\[ |\mathbf{b}| = \sqrt{(2)^2 + (2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

4. **Calculate the Unit Vectors**:
\[ \hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|} = \frac{2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}}{7} \]
\[ \hat{\mathbf{b}} = \frac{\mathbf{b}}{|\mathbf{b}|} = \frac{2\mathbf{i} + 2\mathbf{j} - \mathbf{k}}{3} \]

5. **Calculate the Projection of \( \mathbf{b} \) on \( \mathbf{a} \)**:
\[ \text{Projection of } \mathbf{b} \text{ on } \mathbf{a} = \frac{\mathbf{b} \cdot \hat{\mathbf{a}}}{|\hat{\mathbf{a}}|} = \mathbf{b} \cdot \hat{\mathbf{a}} = \frac{\mathbf{b} \cdot \mathbf{a}}{|\mathbf{a}|} \]
\[ \mathbf{b} \cdot \mathbf{a} = (2)(2) + (2)(-3) + (-1)(6) = 4 - 6 - 6 = -8 \]
Thus,
\[ \text{Projection of } \mathbf{b} \text{ on } \mathbf{a} = \frac{-8}{7} \]

6. **Calculate the Projection of \( \mathbf{a} \) on \( \mathbf{b} \)**:
\[ \text{Projection of } \mathbf{a} \text{ on } \mathbf{b} = \frac{\mathbf{a} \cdot \hat{\mathbf{b}}}{|\hat{\mathbf{b}}|} = \mathbf{a} \cdot \hat{\mathbf{b}} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]
\[ \mathbf{a} \cdot \mathbf{b} = (2)(2) + (-3)(2) + (6)(-1) = 4 - 6 - 6 = -8 \]
Thus,
\[ \text{Projection of } \mathbf{a} \text{ on } \mathbf{b} = \frac{-8}{3} \]

7. **Calculate \( p \)**:
\[ p = \frac{\text{Projection of } \mathbf{b} \text{ on } \mathbf{a}}{\text{Projection of } \mathbf{a} \text{ on } \mathbf{b}} = \frac{\frac{-8}{7}}{\frac{-8}{3}} = \frac{-8}{7} \cdot \frac{3}{-8} = \frac{3}{7} \]

### Final Answer
\[ p = \frac{3}{7} \]