Projection of the vector `2hat(i) + 3hat(j) + 2hat(k)` on the vector `hat(i) - 2hat(j) + 3hat(k)` is :

Options

A. `(2)/(sqrt(14))`
B. `(1)/(sqrt(14))`
C. `(3)/(sqrt(17))`
D. `(3)/(sqrt(14))`

Answer

To find the projection of the vector \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 2\hat{k} \) on the vector \( \mathbf{B} = \hat{i} - 2\hat{j} + 3\hat{k} \), we can follow these steps:

### Step 1: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)

The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:

\[ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z \]

For our vectors:

- \( A_x = 2, A_y = 3, A_z = 2 \)
- \( B_x = 1, B_y = -2, B_z = 3 \)

Now, substituting the values:

\[ \mathbf{A} \cdot \mathbf{B} = (2)(1) + (3)(-2) + (2)(3) \]

Calculating this gives:

\[ = 2 - 6 + 6 = 2 \]

### Step 2: Calculate the magnitude of vector \( \mathbf{B} \)

The magnitude of vector \( \mathbf{B} \) is calculated using the formula:

\[ |\mathbf{B}| = \sqrt{B_x^2 + B_y^2 + B_z^2} \]

Substituting the values:

\[ |\mathbf{B}| = \sqrt{(1)^2 + (-2)^2 + (3)^2} \]

Calculating this gives:

\[ = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 3: Calculate the projection of \( \mathbf{A} \) on \( \mathbf{B} \)

The projection of vector \( \mathbf{A} \) on vector \( \mathbf{B} \) is given by the formula:

\[ \text{Projection of } \mathbf{A} \text{ on } \mathbf{B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} \]

Substituting the values we calculated:

\[ \text{Projection of } \mathbf{A} \text{ on } \mathbf{B} = \frac{2}{\sqrt{14}} \]

### Final Answer

Thus, the projection of the vector \( 2\hat{i} + 3\hat{j} + 2\hat{k} \) on the vector \( \hat{i} - 2\hat{j} + 3\hat{k} \) is:

\[ \frac{2}{\sqrt{14}} \]