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 use the formula for the projection of vector \( \mathbf{A} \) onto vector \( \mathbf{B} \):

\[ \text{Projection of } \mathbf{A} \text{ on } \mathbf{B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2} \mathbf{B} \]

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

The dot product of two vectors \( \mathbf{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \mathbf{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) is given by:

\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]

For our vectors:

- \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 2\hat{k} \) (where \( a_1 = 2, a_2 = 3, a_3 = 2 \))
- \( \mathbf{B} = \hat{i} - 2\hat{j} + 3\hat{k} \) (where \( b_1 = 1, b_2 = -2, b_3 = 3 \))

Calculating the dot product:

\[ \mathbf{A} \cdot \mathbf{B} = (2)(1) + (3)(-2) + (2)(3) = 2 - 6 + 6 = 2 \]

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

The magnitude of vector \( \mathbf{B} \) is given by:

\[ |\mathbf{B}| = \sqrt{b_1^2 + b_2^2 + b_3^2} \]

Calculating \( |\mathbf{B}|^2 \):

\[ |\mathbf{B}|^2 = 1^2 + (-2)^2 + 3^2 = 1 + 4 + 9 = 14 \]

### Step 3: Substitute into the projection formula

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

\[ \text{Projection of } \mathbf{A} \text{ on } \mathbf{B} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2} \mathbf{B} = \frac{2}{14} \mathbf{B} = \frac{1}{7} \mathbf{B} \]

### Step 4: Write the final answer

Substituting \( \mathbf{B} \):

\[ \text{Projection of } \mathbf{A} \text{ on } \mathbf{B} = \frac{1}{7} (\hat{i} - 2\hat{j} + 3\hat{k}) = \frac{1}{7}\hat{i} - \frac{2}{7}\hat{j} + \frac{3}{7}\hat{k} \]

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:

\[ \frac{1}{7}\hat{i} - \frac{2}{7}\hat{j} + \frac{3}{7}\hat{k} \]