If `vecA=2hati + 3hatj` and `vecB= 5hati + 7hatj`, then the vector having the magnitude of vecA and direction of vecB would be

Options

A. `(74/13)^(1/2) (2hati+3hatj)`
B. `sqrt(74(5hati+7hatj))`
C. `(13/74)^(1/2) (5hati+7hatj)`
D. `5sqrt13hati+7sqrt13hatj`

Answer

To find the vector that has the magnitude of vector A and the direction of vector B, we can follow these steps:

### Step 1: Determine the Magnitude of Vector A
Given:
\[ \vec{A} = 2\hat{i} + 3\hat{j} \]
The magnitude of vector A is calculated using the formula:
\[ |\vec{A}| = \sqrt{(A_x)^2 + (A_y)^2} \]
Substituting the values:
\[ |\vec{A}| = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 2: Determine the Magnitude of Vector B
Given:
\[ \vec{B} = 5\hat{i} + 7\hat{j} \]
The magnitude of vector B is calculated similarly:
\[ |\vec{B}| = \sqrt{(B_x)^2 + (B_y)^2} \]
Substituting the values:
\[ |\vec{B}| = \sqrt{(5)^2 + (7)^2} = \sqrt{25 + 49} = \sqrt{74} \]

### Step 3: Find the Unit Vector in the Direction of Vector B
The unit vector in the direction of vector B, denoted as \(\hat{B}\), is given by:
\[ \hat{B} = \frac{\vec{B}}{|\vec{B}|} \]
Substituting the values:
\[ \hat{B} = \frac{5\hat{i} + 7\hat{j}}{\sqrt{74}} \]

### Step 4: Construct the New Vector with Magnitude of A and Direction of B
Let \(\vec{X}\) be the vector that has the magnitude of vector A and the direction of vector B. We can express \(\vec{X}\) as:
\[ \vec{X} = |\vec{A}| \cdot \hat{B} \]
Substituting the values we found:
\[ \vec{X} = \sqrt{13} \cdot \frac{5\hat{i} + 7\hat{j}}{\sqrt{74}} \]

### Step 5: Simplify the Expression
\[ \vec{X} = \frac{\sqrt{13}}{\sqrt{74}}(5\hat{i} + 7\hat{j}) \]
This can also be expressed as:
\[ \vec{X} = \frac{5\sqrt{13}}{\sqrt{74}}\hat{i} + \frac{7\sqrt{13}}{\sqrt{74}}\hat{j} \]

### Final Result
The vector having the magnitude of \(\vec{A}\) and the direction of \(\vec{B}\) is:
\[ \vec{X} = \frac{5\sqrt{13}}{\sqrt{74}}\hat{i} + \frac{7\sqrt{13}}{\sqrt{74}}\hat{j} \]