What is the angle between `vec(A)` and `vec(B)`, If `vec(A)` and `vec(B)` are the adjacent sides of a parallelogram drwan from a common point and the area of the parallelogram is `AB//2`?

Options

A. `15^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Explanation

To find the angle between vectors \(\vec{A}\) and \(\vec{B}\) given that they are the adjacent sides of a parallelogram and the area of the parallelogram is \(\frac{AB}{2}\), we can follow these steps:

### Step 1: Understand the formula for the area of a parallelogram
The area \(A\) of a parallelogram formed by two vectors \(\vec{A}\) and \(\vec{B}\) can be expressed using the cross product:
\[ \text{Area} = |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]
where \(\theta\) is the angle between the vectors \(\vec{A}\) and \(\vec{B}\).

### Step 2: Set up the equation with the given area
According to the problem, the area of the parallelogram is given as:
\[ \text{Area} = \frac{AB}{2} \]
where \(A = |\vec{A}|\) and \(B = |\vec{B}|\).

### Step 3: Equate the two expressions for area
We can set the two expressions for the area equal to each other:
\[ |\vec{A}| |\vec{B}| \sin \theta = \frac{AB}{2} \]

### Step 4: Simplify the equation
Since \(A = |\vec{A}|\) and \(B = |\vec{B}|\), we can substitute these into the equation:
\[ AB \sin \theta = \frac{AB}{2} \]
Now, divide both sides by \(AB\) (assuming \(A\) and \(B\) are not zero):
\[ \sin \theta = \frac{1}{2} \]

### Step 5: Solve for the angle \(\theta\)
The angle \(\theta\) that satisfies \(\sin \theta = \frac{1}{2}\) is:
\[ \theta = 30^\circ \]

### Conclusion
Thus, the angle between vectors \(\vec{A}\) and \(\vec{B}\) is \(30^\circ\).

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