If vectors `a=2hat(i)-3hat(j)+6hat(k)` and vector `b=-2hat(i)+2hat(j)-hat(k)`, then (projection of vector a on b vectors)/(projection of vector b on a vector) is equal to

Options

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

Answer

To solve the problem, we need to find the ratio of the projection of vector **a** on vector **b** to the projection of vector **b** on vector **a**.

Given:
- **a** = \(2\hat{i} - 3\hat{j} + 6\hat{k}\)
- **b** = \(-2\hat{i} + 2\hat{j} - \hat{k}\)

### Step 1: Calculate the dot product of vectors **a** and **b**.

The dot product of two vectors **a** and **b** is given by:
\[ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z \]
For our vectors:
\[ \mathbf{a} \cdot \mathbf{b} = (2)(-2) + (-3)(2) + (6)(-1) = -4 - 6 - 6 = -16 \]

### Step 2: Calculate the magnitude of vector **a**.

The magnitude of vector **a** is given by:
\[ |\mathbf{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2} \]
Calculating for vector **a**:
\[ |\mathbf{a}| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

### Step 3: Calculate the magnitude of vector **b**.

The magnitude of vector **b** is given by:
\[ |\mathbf{b}| = \sqrt{b_x^2 + b_y^2 + b_z^2} \]
Calculating for vector **b**:
\[ |\mathbf{b}| = \sqrt{(-2)^2 + 2^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 4: Calculate the projection of vector **a** on vector **b**.

The projection of vector **a** on vector **b** is given by:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \]
We need the ratio of projections:
\[ \frac{\text{proj}_{\mathbf{b}} \mathbf{a}}{\text{proj}_{\mathbf{a}} \mathbf{b}} = \frac{\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2}}{\frac{\mathbf{b} \cdot \mathbf{a}}{|\mathbf{a}|^2}} \]
Since \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\), we can simplify:
\[ \frac{\text{proj}_{\mathbf{b}} \mathbf{a}}{\text{proj}_{\mathbf{a}} \mathbf{b}} = \frac{|\mathbf{a}|^2}{|\mathbf{b}|^2} \]

### Step 5: Substitute the magnitudes into the ratio.

Now substituting the magnitudes we calculated:
\[ \frac{|\mathbf{a}|^2}{|\mathbf{b}|^2} = \frac{7^2}{3^2} = \frac{49}{9} \]

### Final Answer

Thus, the ratio of the projection of vector **a** on vector **b** to the projection of vector **b** on vector **a** is:
\[ \frac{49}{9} \]