Question

What may the cross product of two vectors be used for?

• A. Area of rectangle
• B. Area of square
• C. Area of parallelogram
• D. Perimeter of rectangle

Hint:

For two vectors \( \vec{A} \) and \( \vec{B} \), the magnitude of the cross product \( |\vec{A} \times \vec{B}| \) gives the area of the parallelogram formed by the vectors.

Answer 1

The Correct Option is C

Step 1: Understanding the Cross Product
The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) results in a vector that is perpendicular to both vectors.
The magnitude of this vector is given by: \[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \] Where \( \theta \) is the angle between the two vectors.
Step 2: Geometrical Interpretation
The magnitude of the cross product represents the area of the parallelogram formed by the two vectors.
This is because the area of a parallelogram is given by \( \text{Area} = \text{Base} \times \text{Height} \), which can be computed as \( |\vec{A}| |\vec{B}| \sin \theta \), where \( \sin \theta \) gives the perpendicular height of the parallelogram when \( |\vec{A}| \) and \( |\vec{B}| \) are the sides of the parallelogram.
Step 3: Conclusion Thus, the cross product is used to calculate the area of a parallelogram formed by two vectors.