Given `vec(A) = 2hat(i) + 3hat(j) and vec(B) = hat(i) + hat(j)` . What is the vector component of `vec(A)` in the direction of `vec(B)` ?

Explanation

To find the vector component of \(\vec{A}\) in the direction of \(\vec{B}\), we can follow these steps:

### Step 1: Identify the vectors
Given:
\[ \vec{A} = 2\hat{i} + 3\hat{j} \]
\[ \vec{B} = \hat{i} + \hat{j} \]

### Step 2: Calculate the magnitude of vector \(\vec{B}\)
The magnitude of \(\vec{B}\) is calculated as follows:
\[ |\vec{B}| = \sqrt{(1^2 + 1^2)} = \sqrt{2} \]

### Step 3: Calculate the dot product \(\vec{A} \cdot \vec{B}\)
The dot product of \(\vec{A}\) and \(\vec{B}\) is given by:
\[ \vec{A} \cdot \vec{B} = (2\hat{i} + 3\hat{j}) \cdot (\hat{i} + \hat{j}) = 2 \cdot 1 + 3 \cdot 1 = 2 + 3 = 5 \]

### Step 4: Calculate the component of \(\vec{A}\) in the direction of \(\vec{B}\)
The component of \(\vec{A}\) in the direction of \(\vec{B}\) is given by:
\[ \text{Component of } \vec{A} \text{ along } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} \]
Substituting the values we calculated:
\[ \text{Component of } \vec{A} \text{ along } \vec{B} = \frac{5}{\sqrt{2}} \]

### Step 5: Find the unit vector in the direction of \(\vec{B}\)
The unit vector \(\hat{b}\) in the direction of \(\vec{B}\) is:
\[ \hat{b} = \frac{\vec{B}}{|\vec{B}|} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} \]

### Step 6: Calculate the vector component of \(\vec{A}\) in the direction of \(\vec{B}\)
Now, we multiply the magnitude of the component by the unit vector:
\[ \text{Vector component of } \vec{A} \text{ along } \vec{B} = \left(\frac{5}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}\right) \]
\[ = \frac{5}{2} \hat{i} + \frac{5}{2} \hat{j} \]

### Final Answer
Thus, the vector component of \(\vec{A}\) in the direction of \(\vec{B}\) is:
\[ \frac{5}{2} \hat{i} + \frac{5}{2} \hat{j} \]