1. What is the Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space that results in a vector perpendicular to both original vectors. Its magnitude equals the area of the parallelogram spanned by the two vectors.

2. How Does the Calculator Work?

The calculator uses the cross product magnitude formula:

Where:

• \( |\vec{a}| \) — Magnitude of vector A
• \( |\vec{b}| \) — Magnitude of vector B
• \( \theta \) — Angle between vectors A and B (in degrees)
• \( \sin(\theta) \) — Sine of the angle between vectors

Explanation: The cross product magnitude represents the area of the parallelogram formed by the two vectors and indicates how perpendicular the vectors are to each other.

3. Importance of Cross Product Calculation

Details: Cross products are essential in physics for calculating torque, angular momentum, and magnetic force. In engineering, they're used for moment calculations and 3D graphics transformations.

4. Using the Calculator

Tips: Enter the magnitudes of both vectors (must be positive values) and the angle between them (0-180 degrees). The calculator will compute the cross product magnitude.

5. Frequently Asked Questions (FAQ)

Q1: What does a cross product of zero mean?
A: A cross product magnitude of zero indicates that the vectors are parallel (angle = 0° or 180°) or one/both vectors have zero magnitude.

Q2: How is this different from dot product?
A: Dot product gives a scalar result representing projection, while cross product gives a vector result representing perpendicularity and area.

Q3: What's the maximum possible cross product value?
A: The maximum occurs when vectors are perpendicular (θ = 90°), where sin(90°) = 1, giving maximum magnitude of |a| × |b|.

Q4: Can cross product be negative?
A: The magnitude is always non-negative, but the direction vector can indicate positive or negative orientation in 3D space.

Q5: What units does the result have?
A: The result has units squared (units²) since it represents an area measurement.