Find the vector projection of the vector :
`2hat(i)-hat(j)+hat(k)` on `hat(i)-2hat(j)+hat(k)`.

Answer

To find the vector projection of the vector \( \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \) on the vector \( \mathbf{b} = \hat{i} - 2\hat{j} + \hat{k} \), we can use the formula for vector projection:

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \]

### Step 1: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \)

\[ \mathbf{a} \cdot \mathbf{b} = (2\hat{i} - \hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} + \hat{k}) \]

Calculating the dot product:

\[ = 2 \cdot 1 + (-1) \cdot (-2) + 1 \cdot 1 \]
\[ = 2 + 2 + 1 = 5 \]

### Step 2: Calculate the dot product \( \mathbf{b} \cdot \mathbf{b} \)

\[ \mathbf{b} \cdot \mathbf{b} = (\hat{i} - 2\hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} + \hat{k}) \]

Calculating the dot product:

\[ = 1 \cdot 1 + (-2) \cdot (-2) + 1 \cdot 1 \]
\[ = 1 + 4 + 1 = 6 \]

### Step 3: Use the projection formula

Now, substitute the values into the projection formula:

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{5}{6} \mathbf{b} \]

### Step 4: Substitute \( \mathbf{b} \)

Substituting \( \mathbf{b} = \hat{i} - 2\hat{j} + \hat{k} \):

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{5}{6} (\hat{i} - 2\hat{j} + \hat{k}) \]

### Step 5: Distribute \( \frac{5}{6} \)

Distributing \( \frac{5}{6} \):

\[ = \frac{5}{6}\hat{i} - \frac{10}{6}\hat{j} + \frac{5}{6}\hat{k} \]

### Final Answer

Thus, the vector projection of \( \mathbf{a} \) on \( \mathbf{b} \) is:

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{5}{6}\hat{i} - \frac{5}{3}\hat{j} + \frac{5}{6}\hat{k} \]