Find the components of a vector `A = 2hati + 3hatj` along the directions of `hati + hatj and hati - hatj.`

Explanation

To find the components of the vector \( \mathbf{A} = 2\hat{i} + 3\hat{j} \) along the directions of \( \hat{i} + \hat{j} \) and \( \hat{i} - \hat{j} \), we will follow these steps:

### Step 1: Define the vectors
Let:
- \( \mathbf{A} = 2\hat{i} + 3\hat{j} \)
- \( \mathbf{r_1} = \hat{i} + \hat{j} \)
- \( \mathbf{r_2} = \hat{i} - \hat{j} \)

### Step 2: Calculate the magnitude of \( \mathbf{r_1} \) and \( \mathbf{r_2} \)
The magnitude of \( \mathbf{r_1} \):
\[ |\mathbf{r_1}| = \sqrt{(1^2 + 1^2)} = \sqrt{2} \]

The magnitude of \( \mathbf{r_2} \):
\[ |\mathbf{r_2}| = \sqrt{(1^2 + (-1)^2)} = \sqrt{2} \]

### Step 3: Find the component of \( \mathbf{A} \) along \( \mathbf{r_1} \)
The component of \( \mathbf{A} \) along \( \mathbf{r_1} \) is given by:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{r_1} = \frac{\mathbf{A} \cdot \mathbf{r_1}}{|\mathbf{r_1}|} \]

Calculating the dot product \( \mathbf{A} \cdot \mathbf{r_1} \):
\[ \mathbf{A} \cdot \mathbf{r_1} = (2\hat{i} + 3\hat{j}) \cdot (\hat{i} + \hat{j}) = 2 \cdot 1 + 3 \cdot 1 = 2 + 3 = 5 \]

Now, substituting into the component formula:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{r_1} = \frac{5}{\sqrt{2}} = \frac{5}{\sqrt{2}} \]

### Step 4: Find the component of \( \mathbf{A} \) along \( \mathbf{r_2} \)
The component of \( \mathbf{A} \) along \( \mathbf{r_2} \) is given by:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{r_2} = \frac{\mathbf{A} \cdot \mathbf{r_2}}{|\mathbf{r_2}|} \]

Calculating the dot product \( \mathbf{A} \cdot \mathbf{r_2} \):
\[ \mathbf{A} \cdot \mathbf{r_2} = (2\hat{i} + 3\hat{j}) \cdot (\hat{i} - \hat{j}) = 2 \cdot 1 + 3 \cdot (-1) = 2 - 3 = -1 \]

Now, substituting into the component formula:
\[ \text{Component of } \mathbf{A} \text{ along } \mathbf{r_2} = \frac{-1}{\sqrt{2}} = -\frac{1}{\sqrt{2}} \]

### Final Result
The components of the vector \( \mathbf{A} \) are:
- Along \( \hat{i} + \hat{j} \): \( \frac{5}{\sqrt{2}} \)
- Along \( \hat{i} - \hat{j} \): \( -\frac{1}{\sqrt{2}} \)