The vector projection of a vector `3hat(i)+4hat(k)` on y-axis is

Options

A. 5
B. 4
C. 3
D. Zero

Explanation

To find the vector projection of the vector \( \mathbf{A} = 3\hat{i} + 4\hat{k} \) on the y-axis, we can follow these steps:

### Step 1: Identify the vectors
We have:
- Vector \( \mathbf{A} = 3\hat{i} + 4\hat{k} \)
- The unit vector along the y-axis \( \mathbf{B} = \hat{j} \)

### Step 2: Use the projection formula
The formula for the vector projection of \( \mathbf{A} \) onto \( \mathbf{B} \) is given by:
\[ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2} \mathbf{B} \]

### Step 3: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)
Calculating the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (3\hat{i} + 4\hat{k}) \cdot \hat{j} = 3(\hat{i} \cdot \hat{j}) + 4(\hat{k} \cdot \hat{j}) \]
Since \( \hat{i} \cdot \hat{j} = 0 \) and \( \hat{k} \cdot \hat{j} = 0 \):
\[ \mathbf{A} \cdot \mathbf{B} = 0 \]

### Step 4: Calculate the magnitude of \( \mathbf{B} \)
The magnitude of \( \mathbf{B} \) (which is the unit vector \( \hat{j} \)):
\[ |\mathbf{B}| = 1 \]
Thus, \( |\mathbf{B}|^2 = 1^2 = 1 \).

### Step 5: Substitute into the projection formula
Substituting the values into the projection formula:
\[ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{0}{1} \hat{j} = 0 \hat{j} \]

### Conclusion
The vector projection of \( \mathbf{A} = 3\hat{i} + 4\hat{k} \) on the y-axis is:
\[ \text{proj}_{\mathbf{B}} \mathbf{A} = 0 \]

### Final Answer
The vector projection of \( 3\hat{i} + 4\hat{k} \) on the y-axis is \( 0 \).

---