Find `vec(a) * vec(b) if |vec(a)| = 6, |vec(b)| = 4 and |vec(a) xx vec(b)| = 12 `

Answer

To find the dot product \( \vec{a} \cdot \vec{b} \) given the magnitudes of the vectors and the magnitude of their cross product, we can follow these steps:

### Step 1: Write down the given information
We have:
- \( |\vec{a}| = 6 \)
- \( |\vec{b}| = 4 \)
- \( |\vec{a} \times \vec{b}| = 12 \)

### Step 2: Use the formula for the magnitude of the cross product
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.

### Step 3: Substitute the known values into the equation
Substituting the values we have:
\[ 12 = 6 \cdot 4 \cdot \sin \theta \]

### Step 4: Simplify the equation
This simplifies to:
\[ 12 = 24 \sin \theta \]
Now, divide both sides by 24:
\[ \sin \theta = \frac{12}{24} = \frac{1}{2} \]

### Step 5: Find the angle \( \theta \)
The angle \( \theta \) for which \( \sin \theta = \frac{1}{2} \) is:
\[ \theta = 30^\circ \]

### Step 6: Use the formula for the dot product
The dot product of two vectors can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

### Step 7: Calculate \( \cos \theta \)
We know:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

### Step 8: Substitute the values into the dot product formula
Now substituting the values into the dot product formula:
\[ \vec{a} \cdot \vec{b} = 6 \cdot 4 \cdot \cos 30^\circ \]
\[ \vec{a} \cdot \vec{b} = 6 \cdot 4 \cdot \frac{\sqrt{3}}{2} \]

### Step 9: Simplify the expression
Calculating this gives:
\[ \vec{a} \cdot \vec{b} = 24 \cdot \frac{\sqrt{3}}{2} = 12\sqrt{3} \]

### Final Answer
Thus, the dot product \( \vec{a} \cdot \vec{b} \) is:
\[ \vec{a} \cdot \vec{b} = 12\sqrt{3} \]

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