If `vec(A)` and `vec(B)` are two such vectors that `|vec(A)| = 2, |vec(B)| = 7` and `vec(A) xx vec(B) = 3 hati +2 hatj + 6 hatk`, 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 properties of the cross product. The steps to solve the problem are as follows:

### Step 1: Understand the given information
We know:
- Magnitude of vector \(\vec{A}\): \(|\vec{A}| = 2\)
- Magnitude of vector \(\vec{B}\): \(|\vec{B}| = 7\)
- Cross product of vectors \(\vec{A}\) and \(\vec{B}\): \(\vec{A} \times \vec{B} = 3\hat{i} + 2\hat{j} + 6\hat{k}\)

### Step 2: Find the magnitude of the cross product
To find the magnitude of the cross product \(|\vec{A} \times \vec{B}|\), we calculate:
\[ |\vec{A} \times \vec{B}| = \sqrt{(3)^2 + (2)^2 + (6)^2} \]
Calculating the squares:
\[ = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]

### Step 3: Use the formula for the magnitude of the cross product
The magnitude of the cross product can also be expressed as:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin \theta \]
Substituting the known values:
\[ 7 = 2 \cdot 7 \cdot \sin \theta \]

### Step 4: Solve for \(\sin \theta\)
Rearranging the equation gives:
\[ \sin \theta = \frac{7}{2 \cdot 7} = \frac{1}{2} \]

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

### Final Answer
The angle between the vectors \(\vec{A}\) and \(\vec{B}\) is \(30^\circ\).

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