Let `|oversettoA_(1)|=3|oversettoA_(2)=5` and `|oversettoA_(1)+oversettoA_(2)|=5`. The value of `(2oversettoA_(1)+3oversettoA_(2))*(3oversettoA_(1)-2oversettoA_(2))`is

Options

A. `-106.5`
B. `-99.5`
C. `-112.5`
D. `-118.5`

Answer

To solve the problem, we need to find the value of \( (2\vec{A_1} + 3\vec{A_2}) \cdot (3\vec{A_1} - 2\vec{A_2}) \) given the magnitudes of the vectors and their sum.

### Step 1: Given Information
We have:
- \( |\vec{A_1}| = 3 \)
- \( |\vec{A_2}| = 5 \)
- \( |\vec{A_1} + \vec{A_2}| = 5 \)

### Step 2: Use the Dot Product Formula
We know that:
\[ |\vec{A_1} + \vec{A_2}|^2 = |\vec{A_1}|^2 + |\vec{A_2}|^2 + 2\vec{A_1} \cdot \vec{A_2} \]
Substituting the known values:
\[ 5^2 = 3^2 + 5^2 + 2\vec{A_1} \cdot \vec{A_2} \]
This simplifies to:
\[ 25 = 9 + 25 + 2\vec{A_1} \cdot \vec{A_2} \]
\[ 25 = 34 + 2\vec{A_1} \cdot \vec{A_2} \]
Rearranging gives:
\[ 2\vec{A_1} \cdot \vec{A_2} = 25 - 34 = -9 \]
Thus,
\[ \vec{A_1} \cdot \vec{A_2} = -\frac{9}{2} \]

### Step 3: Expand the Expression
Now we need to calculate:
\[ (2\vec{A_1} + 3\vec{A_2}) \cdot (3\vec{A_1} - 2\vec{A_2}) \]
Expanding this using the distributive property:
\[ = 2\vec{A_1} \cdot 3\vec{A_1} + 2\vec{A_1} \cdot (-2\vec{A_2}) + 3\vec{A_2} \cdot 3\vec{A_1} + 3\vec{A_2} \cdot (-2\vec{A_2}) \]
This simplifies to:
\[ = 6\vec{A_1} \cdot \vec{A_1} - 4\vec{A_1} \cdot \vec{A_2} + 9\vec{A_2} \cdot \vec{A_1} - 6\vec{A_2} \cdot \vec{A_2} \]

### Step 4: Substitute Values
Now substituting the known values:
- \( \vec{A_1} \cdot \vec{A_1} = |\vec{A_1}|^2 = 3^2 = 9 \)
- \( \vec{A_2} \cdot \vec{A_2} = |\vec{A_2}|^2 = 5^2 = 25 \)
- \( \vec{A_1} \cdot \vec{A_2} = -\frac{9}{2} \)

Substituting these into the expression:
\[ = 6(9) - 4\left(-\frac{9}{2}\right) + 9\left(-\frac{9}{2}\right) - 6(25) \]
Calculating each term:
\[ = 54 + 18 - \frac{81}{2} - 150 \]
Converting \( 54 \) and \( 150 \) to halves for easier calculation:
\[ = \frac{108}{2} + \frac{36}{2} - \frac{81}{2} - \frac{300}{2} \]
Combining these:
\[ = \frac{108 + 36 - 81 - 300}{2} = \frac{-237}{2} = -118.5 \]

### Final Answer
Thus, the value of \( (2\vec{A_1} + 3\vec{A_2}) \cdot (3\vec{A_1} - 2\vec{A_2}) \) is:
\[ \boxed{-118.5} \]