If `vec(b)=3hat(i)+4hat(j)` and `vec(a)=hat(i)-hat(j)` the vector having the same magnitude as that of `vec(b)` and parallel to `vec(a)` is

Options

A. `5/sqrt(2)(hat(i)-hat(j))`
B. `5/sqrt(2)(hat(i)+hat(j))`
C. `5(hat(i)-hat(j))`
D. `5(hat(i)+hat(j))`

Explanation

To solve the problem, we need to find a vector \( \vec{C} \) that has the same magnitude as \( \vec{B} \) and is parallel to \( \vec{A} \).

### Step 1: Find the magnitude of vector \( \vec{B} \)

Given:
\[ \vec{B} = 3\hat{i} + 4\hat{j} \]

The magnitude of \( \vec{B} \) is calculated using the formula:
\[ |\vec{B}| = \sqrt{(3)^2 + (4)^2} \]
Calculating this gives:
\[ |\vec{B}| = \sqrt{9 + 16} = \sqrt{25} = 5 \]

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

Given:
\[ \vec{A} = \hat{i} - \hat{j} \]

The magnitude of \( \vec{A} \) is calculated as:
\[ |\vec{A}| = \sqrt{(1)^2 + (-1)^2} \]
Calculating this gives:
\[ |\vec{A}| = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Find the unit vector of \( \vec{A} \)

The unit vector \( \hat{a} \) in the direction of \( \vec{A} \) is given by:
\[ \hat{a} = \frac{\vec{A}}{|\vec{A}|} = \frac{\hat{i} - \hat{j}}{\sqrt{2}} \]

### Step 4: Find the vector \( \vec{C} \)

Since \( \vec{C} \) is parallel to \( \vec{A} \) and has the same magnitude as \( \vec{B} \), we can express \( \vec{C} \) as:
\[ \vec{C} = |\vec{C}| \hat{a} \]
Given that \( |\vec{C}| = |\vec{B}| = 5 \), we have:
\[ \vec{C} = 5 \hat{a} = 5 \left( \frac{\hat{i} - \hat{j}}{\sqrt{2}} \right) \]
This simplifies to:
\[ \vec{C} = \frac{5}{\sqrt{2}} \hat{i} - \frac{5}{\sqrt{2}} \hat{j} \]

### Final Answer

Thus, the vector \( \vec{C} \) that has the same magnitude as \( \vec{B} \) and is parallel to \( \vec{A} \) is:
\[ \vec{C} = \frac{5}{\sqrt{2}} \hat{i} - \frac{5}{\sqrt{2}} \hat{j} \]

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