Given `vec(A) = 3hat(i) + 2hat(j) and vec(B) = hat(i) + hat(j).` The component of vector `vec(A)` along vector `vec(B)` is

Options

A. `(1)/(sqrt2)`
B. `(3)/(sqrt2)`
C. `(5)/(sqrt2)`
D. `(7)/(sqrt2)`

Answer

To find the component of vector \(\vec{A}\) along vector \(\vec{B}\), we can use the formula for the component of one vector along another. The component of vector \(\vec{A}\) along vector \(\vec{B}\) can be calculated using the following formula:

\[ \text{Component of } \vec{A} \text{ along } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} \]

### Step 1: Find the dot product \(\vec{A} \cdot \vec{B}\)

Given:
\[ \vec{A} = 3\hat{i} + 2\hat{j} \]
\[ \vec{B} = \hat{i} + \hat{j} \]

The dot product \(\vec{A} \cdot \vec{B}\) is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (3\hat{i} + 2\hat{j}) \cdot (\hat{i} + \hat{j}) = 3(\hat{i} \cdot \hat{i}) + 2(\hat{j} \cdot \hat{j}) + 3(\hat{i} \cdot \hat{j}) + 2(\hat{j} \cdot \hat{i}) \]
Since \(\hat{i} \cdot \hat{i} = 1\), \(\hat{j} \cdot \hat{j} = 1\), and \(\hat{i} \cdot \hat{j} = 0\):
\[ \vec{A} \cdot \vec{B} = 3(1) + 2(1) + 0 + 0 = 3 + 2 = 5 \]

### Step 2: Find the magnitude of vector \(\vec{B}\)

The magnitude of vector \(\vec{B}\) is calculated as follows:
\[ |\vec{B}| = \sqrt{(\hat{i}^2 + \hat{j}^2)} = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the component of \(\vec{A}\) along \(\vec{B}\)

Now we can substitute the values into the component formula:
\[ \text{Component of } \vec{A} \text{ along } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} = \frac{5}{\sqrt{2}} \]

### Final Answer

The component of vector \(\vec{A}\) along vector \(\vec{B}\) is:
\[ \frac{5}{\sqrt{2}} \]