The projection of the vector `hat(i) +hat(j) + hat(k)` along vector `hat(j)` is

Options

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

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

### Step 3: Calculate the dot product \(\mathbf{A} \cdot \mathbf{B}\)
To find \(\mathbf{A} \cdot \mathbf{B}\):
\[ \mathbf{A} \cdot \mathbf{B} = (\hat{i} + \hat{j} + \hat{k}) \cdot \hat{j} \]
Calculating the dot product:
\[ \hat{i} \cdot \hat{j} = 0, \quad \hat{j} \cdot \hat{j} = 1, \quad \hat{k} \cdot \hat{j} = 0 \]
Thus,
\[ \mathbf{A} \cdot \mathbf{B} = 0 + 1 + 0 = 1 \]

### Step 4: Calculate the magnitude of \(\mathbf{B}\)
The magnitude of \(\mathbf{B}\) is:
\[ |\mathbf{B}| = |\hat{j}| = 1 \]

### Step 5: Substitute into the projection formula
Now substitute the values into the projection formula:
\[ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{1}{1^2} \mathbf{B} = 1 \cdot \hat{j} = \hat{j} \]

### Conclusion
The projection of the vector \(\hat{i} + \hat{j} + \hat{k}\) along the vector \(\hat{j}\) is:
\[ \hat{j} \]