The projection of the vector `hati+hatj+hatk` along the vector of `hatj` is

Options

A. 1
B. 0
C. 2
D. -1

Answer

To find the projection of the vector \( \hat{i} + \hat{j} + \hat{k} \) along the vector \( \hat{j} \), we can follow these steps:

### Step 1: Define the vectors
Let:
- \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k} \)
- \( \mathbf{b} = \hat{j} \)

### Step 2: Use the projection formula
The formula for the projection of vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) 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 \( \mathbf{a} \cdot \mathbf{b} \)
Now, we need to calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \):

\[ \mathbf{a} \cdot \mathbf{b} = (\hat{i} + \hat{j} + \hat{k}) \cdot \hat{j} \]

Calculating the dot product:

\[ \mathbf{a} \cdot \mathbf{b} = \hat{i} \cdot \hat{j} + \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{j} = 0 + 1 + 0 = 1 \]

### Step 4: Calculate the magnitude of \( \mathbf{b} \)
Next, we calculate the magnitude of \( \mathbf{b} \):

\[ |\mathbf{b}| = |\hat{j}| = 1 \]

So,

\[ |\mathbf{b}|^2 = 1^2 = 1 \]

### Step 5: Substitute into the projection formula
Now substitute the values into the projection formula:

\[ \text{proj}_{\mathbf{b}} \mathbf{a} = \frac{1}{1} \mathbf{b} = 1 \cdot \hat{j} = \hat{j} \]

### Conclusion
Thus, the projection of the vector \( \hat{i} + \hat{j} + \hat{k} \) along the vector \( \hat{j} \) is:

\[ \hat{j} \]