Understanding the Cross Product of Vectors A and B

The cross product is a fundamental operation in the field of vector algebra. It is used to compute a vector that is orthogonal (perpendicular) to the two vectors used in the operation. In this article, we will explore the cross product of vectors A 2i 4j 6k and B 4i - 2j 12k, and explain the process step-by-step.

What is a Cross Product?

A cross product between two vectors A and B produces a vector that is orthogonal to both A and B. The magnitude of the resulting vector is equal to the area of the parallelogram spanned by A and B. Moreover, the direction of the cross product follows the right-hand rule.

Formula for Cross Product

The cross product of two vectors A and B can be calculated using the formula:

A x B a2b3 - a3b2i a3b1 - a1b3j a1b2 - a2b1k

Calculating the Cross Product of Vectors A and B

Let us apply the above formula to find the cross product of vectors A 2i 4j 6k and B 4i - 2j 12k.

Step-by-Step Calculation

Identify the components of vectors A and B: A 2i 4j 6k, where a1 2, a2 4, a3 6 B 4i - 2j 12k, where b1 4, b2 -2, b3 12 Calculate each component of the cross product: (a2b3 - a3b2)i (4 * 12 - 6 * (-2))i (48 12)i 60i (a3b1 - a1b3)j (6 * 4 - 2 * 12)j (24 - 24)j 0j (a1b2 - a2b1)k (2 * (-2) - 4 * 4)k (-4 - 16)k -20k

Result

The cross product of vectors A and B is:

A x B 60i 0j - 20k

Practical Applications of Cross Product

The cross product has numerous applications in physics and engineering. It is particularly useful in:

Mechanics: Calculating torque and angular momentum. Electronics: Understanding the behavior of magnetic fields. Computer Graphics: Generating 3D rotations and transformations.

Conclusion

Understanding the cross product of vectors is essential for anyone working with vector algebra or related fields. This article provides a comprehensive explanation of how to calculate the cross product and its significance in various applications.