Find the vector projection of the vector :
`7hat(i)+hat(j)-hat(k)` on `2hat(i)+6hat(j)+3hat(k)`

Answer

To find the vector projection of the vector \( \mathbf{a} = 7\hat{i} + \hat{j} - \hat{k} \) on the vector \( \mathbf{b} = 2\hat{i} + 6\hat{j} + 3\hat{k} \), we can use the formula for the vector projection of \( \mathbf{a} \) onto \( \mathbf{b} \):

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} \]

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

\[ \mathbf{a} \cdot \mathbf{b} = (7\hat{i} + \hat{j} - \hat{k}) \cdot (2\hat{i} + 6\hat{j} + 3\hat{k}) \]

Calculating the dot product:

\[ = 7 \cdot 2 + 1 \cdot 6 + (-1) \cdot 3 \]
\[ = 14 + 6 - 3 \]
\[ = 17 \]

### Step 2: Calculate the magnitude squared of \( \mathbf{b} \)

\[ |\mathbf{b}|^2 = (2\hat{i} + 6\hat{j} + 3\hat{k}) \cdot (2\hat{i} + 6\hat{j} + 3\hat{k}) \]

Calculating the magnitude squared:

\[ = 2^2 + 6^2 + 3^2 \]
\[ = 4 + 36 + 9 \]
\[ = 49 \]

### Step 3: Substitute into the projection formula

Now we can substitute the values we found into the projection formula:

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{17}{49} \mathbf{b} \]

### Step 4: Write the final answer

Substituting \( \mathbf{b} \):

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{17}{49} (2\hat{i} + 6\hat{j} + 3\hat{k}) \]

Distributing \( \frac{17}{49} \):

\[ = \frac{34}{49} \hat{i} + \frac{102}{49} \hat{j} + \frac{51}{49} \hat{k} \]

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

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{34}{49} \hat{i} + \frac{102}{49} \hat{j} + \frac{51}{49} \hat{k} \]