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

Options

A. `(pi)/(6)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(2pi)/( 3)`

Answer

To find the angle between the vectors \(\vec{a}\) and \(\vec{b}\), we can follow these steps:

### Step 1: Given Information
We have the following information:
- \(|\vec{a}| = 2\)
- \(|\vec{b}| = 7\)
- \(\vec{a} \times \vec{b} = 3\hat{i} + 2\hat{j} + 6\hat{k}\)

### Step 2: Calculate the Magnitude of the Cross Product
We need to find the magnitude of the cross product \(\vec{a} \times \vec{b}\):
\[ |\vec{a} \times \vec{b}| = \sqrt{(3)^2 + (2)^2 + (6)^2} \]
Calculating this gives:
\[ |\vec{a} \times \vec{b}| = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]

### Step 3: Use the Formula for the Magnitude of the Cross Product
The magnitude of the cross product can also be expressed in terms of the magnitudes of the vectors and the sine of the angle \(\theta\) between them:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin \theta \]
Substituting the known values:
\[ 7 = 2 \cdot 7 \cdot \sin \theta \]

### Step 4: Simplify the Equation
Now we simplify the equation:
\[ 7 = 14 \cdot \sin \theta \]
Dividing both sides by 14:
\[ \sin \theta = \frac{7}{14} = \frac{1}{2} \]

### Step 5: Find the Angle \(\theta\)
The angle \(\theta\) for which \(\sin \theta = \frac{1}{2}\) is:
\[ \theta = \frac{\pi}{6} \]

### Conclusion
Thus, the angle between the vectors \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{6}\).

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