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 components of the vector
The given vector is \( \mathbf{A} = 3\hat{i} + 0\hat{j} + 4\hat{k} \). Here, we can see that:
- The x-component (i-component) is 3.
- The y-component (j-component) is 0.
- The z-component (k-component) is 4.

### Step 2: Understand the projection concept
The projection of a vector \( \mathbf{A} \) onto another vector \( \mathbf{B} \) is given by the formula:
\[ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{\mathbf{B} \cdot \mathbf{B}} \mathbf{B} \]
In this case, we want to project \( \mathbf{A} \) onto the y-axis, which can be represented by the unit vector \( \hat{j} \).

### Step 3: Calculate the dot product
The dot product \( \mathbf{A} \cdot \hat{j} \) is calculated as follows:
\[ \mathbf{A} \cdot \hat{j} = (3\hat{i} + 0\hat{j} + 4\hat{k}) \cdot (0\hat{i} + 1\hat{j} + 0\hat{k}) = 3 \cdot 0 + 0 \cdot 1 + 4 \cdot 0 = 0 \]

### Step 4: Calculate the magnitude of \( \hat{j} \)
The dot product \( \hat{j} \cdot \hat{j} \) is:
\[ \hat{j} \cdot \hat{j} = 1 \]

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

### Conclusion
Thus, the vector projection of \( \mathbf{A} = 3\hat{i} + 4\hat{k} \) on the y-axis is \( 0 \hat{j} \), which means it is zero.

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

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