If `|vec(a)| = 2, |vec(b)| = 7` and `vec(a) xx vec(b) = 3hat(i) + 2hat(j) + 6hat(k)`, find the angle between `vec(a)` and `vec(b)`.

Answer

To find the angle between the vectors \(\vec{a}\) and \(\vec{b}\), we can use the relationship between the magnitudes of the cross product and the sine of the angle between the two vectors.

### Step-by-Step Solution:

1. **Given Information**:
- Magnitude of \(\vec{a}\): \(|\vec{a}| = 2\)
- Magnitude of \(\vec{b}\): \(|\vec{b}| = 7\)
- Cross product: \(\vec{a} \times \vec{b} = 3\hat{i} + 2\hat{j} + 6\hat{k}\)

2. **Calculate the Magnitude of the Cross Product**:
\[ |\vec{a} \times \vec{b}| = \sqrt{(3)^2 + (2)^2 + (6)^2} \]
\[ = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]

3. **Use the Formula for the Magnitude of the Cross Product**:
The formula for the magnitude of the cross product is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
Substituting the known values:
\[ 7 = (2)(7) \sin \theta \]

4. **Simplify the Equation**:
\[ 7 = 14 \sin \theta \]
Dividing both sides by 14:
\[ \sin \theta = \frac{7}{14} = \frac{1}{2} \]

5. **Find the Angle \(\theta\)**:
The angle \(\theta\) whose sine is \(\frac{1}{2}\) is:
\[ \theta = 30^\circ \]

### Final Answer:
The angle between \(\vec{a}\) and \(\vec{b}\) is \(30^\circ\).

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