A parallelogram is constructed on `3veca+vecb and veca-4vecb, where |veca|=6 and |vecb|=8` and `veca and vecb` are anti parallel then the length of the longer diagonal is (A) 40 (B) 64 (C) 32 (D) 48

Options

A. 40
B. 64
C. 32
D. 48

Explanation

To solve the problem step by step, we will find the length of the longer diagonal of the parallelogram constructed on the vectors \( \vec{u} = 3\vec{a} + \vec{b} \) and \( \vec{v} = \vec{a} - 4\vec{b} \).

### Step 1: Identify the vectors
We have:
- \( \vec{u} = 3\vec{a} + \vec{b} \)
- \( \vec{v} = \vec{a} - 4\vec{b} \)

### Step 2: Find the diagonal vector
The diagonals of the parallelogram can be found by adding and subtracting the vectors \( \vec{u} \) and \( \vec{v} \). The longer diagonal \( \vec{d_1} \) is given by:
\[ \vec{d_1} = \vec{u} + \vec{v} = (3\vec{a} + \vec{b}) + (\vec{a} - 4\vec{b}) = 3\vec{a} + \vec{b} + \vec{a} - 4\vec{b} = 4\vec{a} - 3\vec{b} \]

### Step 3: Calculate the magnitude of the diagonal vector
To find the length of the diagonal, we need to calculate the magnitude of \( \vec{d_1} \):
\[ |\vec{d_1}| = |4\vec{a} - 3\vec{b}| \]

### Step 4: Use the formula for the magnitude of a vector
The magnitude of a vector \( \vec{d} = x\vec{a} + y\vec{b} \) can be calculated using:
\[ |\vec{d}|^2 = x^2|\vec{a}|^2 + y^2|\vec{b}|^2 + 2xy(\vec{a} \cdot \vec{b}) \]
In our case:
- \( x = 4 \)
- \( y = -3 \)

### Step 5: Substitute the values
Given \( |\vec{a}| = 6 \) and \( |\vec{b}| = 8 \), and since \( \vec{a} \) and \( \vec{b} \) are anti-parallel, the angle \( \theta = \pi \) implies \( \cos(\pi) = -1 \). Therefore, we have:
\[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos(\pi) = 6 \times 8 \times (-1) = -48 \]

Now substituting into the magnitude formula:
\[ |\vec{d_1}|^2 = 4^2 |\vec{a}|^2 + (-3)^2 |\vec{b}|^2 + 2 \cdot 4 \cdot (-3)(\vec{a} \cdot \vec{b}) \]
\[ = 16 \cdot 36 + 9 \cdot 64 + 2 \cdot 4 \cdot (-3)(-48) \]

### Step 6: Calculate each term
Calculating each term:
1. \( 16 \cdot 36 = 576 \)
2. \( 9 \cdot 64 = 576 \)
3. \( 2 \cdot 4 \cdot (-3)(-48) = 2 \cdot 4 \cdot 3 \cdot 48 = 1152 \)

### Step 7: Combine the results
Now combine these results:
\[ |\vec{d_1}|^2 = 576 + 576 + 1152 = 2304 \]

### Step 8: Take the square root
Finally, take the square root to find the length of the diagonal:
\[ |\vec{d_1}| = \sqrt{2304} = 48 \]

### Conclusion
The length of the longer diagonal is \( 48 \).