Cross product, also known as vector product, is a mathematical operation used primarily in three-dimensional space. It takes two input vectors and returns a new vector that is perpendicular to both input vectors. Calculating the cross product has several applications in physics, engineering, and computer graphics. In this article, we will guide you through a step-by-step process for calculating cross product using elementary algebra.

Prerequisites:

To understand this article, you should be familiar with:

1. Basic algebra.

2. Vectors and their components.

3. Dot product.

What is Cross Product?

Cross product is an operation that takes two three-dimensional vectors as input and returns a third vector that is orthogonal to both of the input vectors. The magnitude of the resulting vector is equal to the area of the parallelogram formed by the two input vectors while the direction follows the right-hand rule.

The formula for Cross Product:

The cross product of vectors A=<a1, a2, a3> and B=<b1, b2, b3> can be calculated using the following formula:

A x B = <a2 * b3 – a3 * b2, a3 * b1 – a1 * b3, a1 * b2 – a2 * b1>

Calculating Cross Product Step-by-Step:

Step 1: Write down the given vectors.

Let’s say we have two given vectors A = <4, -5, 6> and B = <7, 8,-9>.

Step 2: Formulate the expression for cross product.

Based on the formula mentioned above,

A x B = <(A_2 * B_3) – (A_3 * B_2), (A_3 * B_1) – (A_1 * B_3), (A_1 * B_2) – (A_2 * B_1)>

Step 3: Calculate the cross product.

Substitute the components of vectors A and B into the expression,

A x B = <(-5 * -9) – (6 * 8), (6 * 7) – (4 * -9), (4 * 8) – (-5 * 7)>

Now compute the values,

A x B = <45 – 48, 42 + 36, 32 + 35>

Finally, simplifying the values will give us the cross product,

A x B = <-3, 78, 67>

Conclusion:

In this article, we discussed the fundamentals of cross product and provided a step-by-step guide on calculating cross product using a simple example. The process involves understanding the formula, substituting vector component values, and then performing algebraic operations to arrive at the resultant vector. With practice over time, you’ll be able to quickly calculate cross products for various problems in physics, engineering, and computer graphics.