Understanding the Vector (Cross) Product of Vectors A and B - Comprehensive Guide

The vector product, also known as the cross product, is a fundamental concept in vector algebra. Given two vectors A 2i 5j and B 3i 4j, the cross product, denoted as A × B, can be found using the determinant of a 3x3 matrix. This guide will walk you through the calculations involved in finding the cross product.

Step-by-Step Calculation

To calculate the cross product of vectors A and B, we can use the following formula:

A × B i(5*0 - 0*4) - j(2*0 - 0*3) k(2*4 - 5*3)

Let's break down this calculation:

Calculating i-component: 5*0 - 0*4 0 Calculating j-component: 2*0 - 0*3 0 Calculating k-component: 2*4 - 5*3 8 - 15 -7

Therefore, the cross product of A and B is:

A × B 0i - 0j - 7k -7k

Explanation of the Cross Product

The cross product of two vectors results in a vector that is perpendicular to both of the original vectors. In this case, the resulting vector is -7k, indicating that the vector is oriented along the z-axis (negative direction).

Visual Representation and Usage

Visualizing the cross product can help in understanding its significance. In a 3D coordinate system, if vector A is parallel to the x-y plane and vector B is parallel to the same plane, their cross product will be perpendicular to this plane. This is illustrated by the equation:

A × B 0i 0j - 7k

Additional Insights

It's important to note that the cross product operation is non-commutative, i.e., A × B ≠ B × A. In this specific case, we assumed you meant to calculate A × B rather than B × A.

Conclusion

In summary, the cross product of vectors A and B is a powerful tool in vector algebra, providing a vector perpendicular to both original vectors. By mastering the cross product, you can solve a wide range of problems in physics, engineering, and mathematics.

Related Keywords

Vector product Cross product Dot product

For further reading, explore more on vector algebra, vector operations, and their applications in various fields.