Find the angle between `vec(a) and vec(b)` if `|vec(a)| = 4, |vec(b)|= 2 sqrt3 and |vec(a) xx vec(b)| = 12`

Answer

To find the angle between the vectors \(\vec{a}\) and \(\vec{b}\), we can use the formula for the magnitude of the cross product of two vectors:

\[ |\vec{a} \times \vec{b}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin \theta \]

Where:
- \(|\vec{a}|\) is the magnitude of vector \(\vec{a}\)
- \(|\vec{b}|\) is the magnitude of vector \(\vec{b}\)
- \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\)

### Step 1: Substitute the known values into the formula

We are given:
- \(|\vec{a}| = 4\)
- \(|\vec{b}| = 2\sqrt{3}\)
- \(|\vec{a} \times \vec{b}| = 12\)

Substituting these values into the formula gives:

\[ 12 = 4 \cdot (2\sqrt{3}) \cdot \sin \theta \]

### Step 2: Simplify the equation

Calculating the right side:

\[ 12 = 8\sqrt{3} \cdot \sin \theta \]

### Step 3: Isolate \(\sin \theta\)

To isolate \(\sin \theta\), divide both sides by \(8\sqrt{3}\):

\[ \sin \theta = \frac{12}{8\sqrt{3}} = \frac{3}{2\sqrt{3}} \]

### Step 4: Rationalize the denominator

To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{3}\):

\[ \sin \theta = \frac{3\sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3}}{2} \]

### Step 5: Find the angle \(\theta\)

The value of \(\theta\) for which \(\sin \theta = \frac{\sqrt{3}}{2}\) is:

\[ \theta = 60^\circ \quad \text{or} \quad \theta = \frac{\pi}{3} \text{ radians} \]

Thus, the angle between the vectors \(\vec{a}\) and \(\vec{b}\) is \(60^\circ\).