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: Identify the vectors
Given:
\[ \vec{A} = 2\hat{i} + 3\hat{j} \]
\[ \vec{B} = 5\hat{i} + 7\hat{j} \]

### Step 2: Calculate the magnitude of vector A
The magnitude of vector A is calculated using the formula:
\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2} \]
where \(A_x\) and \(A_y\) are the components of vector A.

Substituting the values:
\[ |\vec{A}| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Calculate the magnitude of vector B
Next, we calculate the magnitude of vector B using the same formula:
\[ |\vec{B}| = \sqrt{B_x^2 + B_y^2} \]
Substituting the values:
\[ |\vec{B}| = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74} \]

### Step 4: Find the unit vector of vector B
The unit vector in the direction of vector 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 5: Construct the new vector with magnitude of A and direction of B
The new vector \(\vec{C}\) can be expressed as:
\[ \vec{C} = |\vec{A}| \cdot \hat{B} \]
Substituting the values:
\[ \vec{C} = \sqrt{13} \cdot \left(\frac{5\hat{i} + 7\hat{j}}{\sqrt{74}}\right) \]

### Step 6: Simplify the expression for vector C
\[ \vec{C} = \frac{\sqrt{13} \cdot 5\hat{i} + \sqrt{13} \cdot 7\hat{j}}{\sqrt{74}} \]
This can be simplified to:
\[ \vec{C} = \frac{5\sqrt{13}}{\sqrt{74}}\hat{i} + \frac{7\sqrt{13}}{\sqrt{74}}\hat{j} \]

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

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