If `|veca|=8,|vecb| = 3 and |veca xxvecb|=12 ` , find `veca.vecb`

Answer

To solve the problem, we need to find the dot product of vectors \( \vec{a} \) and \( \vec{b} \) given their magnitudes and the magnitude of their cross product.

### Step-by-Step Solution:

1. **Given Information**:
- Magnitude of \( \vec{a} \): \( |\vec{a}| = 8 \)
- Magnitude of \( \vec{b} \): \( |\vec{b}| = 3 \)
- Magnitude of the cross product \( |\vec{a} \times \vec{b}| = 12 \)

2. **Relationship Between Cross Product and Sine**:
The magnitude of the cross product of two vectors can be expressed as:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
where \( \theta \) is the angle between the two vectors.

3. **Substituting Known Values**:
Substitute the known values into the equation:
\[ 12 = 8 \times 3 \times \sin \theta \]

4. **Calculating \( \sin \theta \)**:
Simplifying the equation:
\[ 12 = 24 \sin \theta \]
Dividing both sides by 24:
\[ \sin \theta = \frac{12}{24} = \frac{1}{2} \]

5. **Finding \( \theta \)**:
The angle \( \theta \) can be found using the inverse sine function:
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) \]
This gives us:
\[ \theta = 30^\circ \]

6. **Using the Dot Product Formula**:
The dot product of two vectors can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

7. **Calculating \( \cos \theta \)**:
Since \( \theta = 30^\circ \):
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

8. **Substituting Values into the Dot Product Formula**:
Now substitute the values into the dot product formula:
\[ \vec{a} \cdot \vec{b} = 8 \times 3 \times \cos 30^\circ \]
\[ = 8 \times 3 \times \frac{\sqrt{3}}{2} \]
\[ = 24 \times \frac{\sqrt{3}}{2} \]
\[ = 12\sqrt{3} \]

### Final Answer:
\[ \vec{a} \cdot \vec{b} = 12\sqrt{3} \]