Find the vector components of `veca= 2hati+ 3hatj` along the directions of `hati+hatj`.

Explanation

To find the vector components of \(\vec{a} = 2\hat{i} + 3\hat{j}\) along the direction of \(\hat{i} + \hat{j}\), we can follow these steps:

### Step 1: Define the vectors
Let \(\vec{a} = 2\hat{i} + 3\hat{j}\) and \(\vec{b} = \hat{i} + \hat{j}\).

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

### Step 3: Calculate the dot product of \(\vec{a}\) and \(\vec{b}\)
The dot product \(\vec{a} \cdot \vec{b}\) 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: Find the component of \(\vec{a}\) along \(\vec{b}\)
Using the formula for the component of \(\vec{a}\) along \(\vec{b}\):
\[ \text{Component of } \vec{a} \text{ along } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]
Substituting the values we found:
\[ \text{Component of } \vec{a} \text{ along } \vec{b} = \frac{5}{\sqrt{2}} \]

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

### Step 6: Calculate the vector component of \(\vec{a}\) along \(\vec{b}\)
Now, we can find the vector component of \(\vec{a}\) along \(\vec{b}\) by multiplying 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
The vector component of \(\vec{a}\) along the direction of \(\hat{i} + \hat{j}\) is:
\[ \frac{5}{2} \hat{i} + \frac{5}{2} \hat{j} \]