What is the projection of `vec(a)=(2hat(i)-hat(j)+hat(k)) ` on `vec(b)=(hat(i)-2hat(j)+hat(k))?`

Options

A. `(2)/(sqrt(3))`
B. `(4)/(sqrt(5))`
C. `(5)/(sqrt(6))`
D. none of these

Explanation

To find the projection of vector **a** on vector **b**, we will use the formula for the projection of one vector onto another.

### Step-by-Step Solution:

1. **Identify the Vectors:**
- Let **a** = \(2\hat{i} - \hat{j} + \hat{k}\)
- Let **b** = \(\hat{i} - 2\hat{j} + \hat{k}\)

2. **Calculate the Dot Product \( \mathbf{a} \cdot \mathbf{b} \):**
\[ \mathbf{a} \cdot \mathbf{b} = (2)(1) + (-1)(-2) + (1)(1) \]
\[ = 2 + 2 + 1 = 5 \]

3. **Calculate the Magnitude of Vector \( \mathbf{b} \):**
\[ |\mathbf{b}| = \sqrt{(1)^2 + (-2)^2 + (1)^2} \]
\[ = \sqrt{1 + 4 + 1} = \sqrt{6} \]

4. **Use the Projection Formula:**
The projection of **a** onto **b** is given by:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \]
First, we need \( |\mathbf{b}|^2 \):
\[ |\mathbf{b}|^2 = 6 \]
Now substitute the values:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{5}{6} \mathbf{b} \]
Substitute **b**:
\[ = \frac{5}{6} (\hat{i} - 2\hat{j} + \hat{k}) \]
\[ = \frac{5}{6} \hat{i} - \frac{10}{6} \hat{j} + \frac{5}{6} \hat{k} \]
\[ = \frac{5}{6} \hat{i} - \frac{5}{3} \hat{j} + \frac{5}{6} \hat{k} \]

### Final Answer:
The projection of vector **a** on vector **b** is:
\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{5}{6} \hat{i} - \frac{5}{3} \hat{j} + \frac{5}{6} \hat{k} \]