If `vec(a) and vec(b)` are vectors such that `|vec(a)| = 2, |vec(b)|= 7 and vec(a) xx vec(b) = 3 hat(i) + 2hat(j) + 6hat(k)`, then what is the acute angle between `vec(a) and vec(b)` ?

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

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

### Step-by-Step Solution:

1. **Given Information**:
- Magnitude of vector \(\vec{a}\): \(|\vec{a}| = 2\)
- Magnitude of vector \(\vec{b}\): \(|\vec{b}| = 7\)
- 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 magnitude of the cross product can also be expressed as:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
Substituting the known values:
\[ 7 = (2)(7) \sin \theta \]

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

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

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