Cross Products

Cross products are the results of vectors multiplied together. The magnitude of the Cross product of two vectors is equal to the product of the magnitude of two vectors and the sine of the smaller angle between the two vectors. A multiplication sign represents the cross products. Since the result of multiplying two vectors is also a vector, cross products are also known as vector products. They are always perpendicular to the direction of the two vectors being multiplied. It is essential to have reliable cross products study material to understand operations on vectors better. 

What are cross products?

The multiplication of two vectors is always in a three-dimensional system. The resultant vector is known as a cross product. Its direction is always perpendicular to the two vectors that are multiplied and can be determined by the right-hand thumb rule. Cross products are obtained by a type of vector multiplication. So if two vectors are multiplied, and the resultant vector is in a direction that is perpendicular to the plane in which the two vectors lie, then it is known as a cross product. For example, if two vectors are in the X – Y plane, then their product will be a vector that will be in the direction of the Z-axis, which is perpendicular to the plane of the X and Y axes. 

Cross product of two vectors

Suppose there are three vectors a, b and c, where c is the product of the former two vectors. Then:

• The angle between a and c is always 90 degrees.
• The angle between b and c is always 90 degrees.
• ‘c’ is always in the direction perpendicular to both vectors.
• The magnitude of the cross product is equal to the product of the magnitudes of the vectors and the sine of the smaller angle made by the two vectors being multiplied.
• If the two vectors being multiplied are at 90 degrees, the resultant cross product will be maximum.
• The cross product of two vectors can also be obtained by calculating the product of the parallelograms formed by the two vectors. 

Formula for finding cross products

The cross product between two products is equal to the total area of the parallelogram formed by the vectors. The formula for cross products gives the magnitude of the resultant vector.

The formula for finding the cross products is:

= |A| x |B| sinθ

θ is the smaller angle between vectors A and B.

The right-hand rule

The right-hand method is used to find out the direction of the resultant cross product. The following steps can be used to determine the direction of the cross product:

• Close your fist.
• Open your middle finger in the direction of the first vector.
• Open your thumb to point perpendicular to the two opened fingers.
• The opened thumb is pointing in the direction of the resultant cross product.

 Properties of cross products

It is essential to understand the properties of cross products so that it is easier to work with them. Furthermore, when the properties of cross products are known, it is easier to correct errors. Following are the properties of cross products:

• The magnitude of cross products is equal to the product of magnitudes of vectors and the sine of the smaller angle between the vectors.
• The operation is not commutative.
• The result of the product of a vector and the sum of two vectors is equal to the sum of the products of the two vectors with the first vector.

A╳ (B+C)=A╳B+A╳C

• The result of the multiplication of a vector with a zero vector is zero.
• If a vector is multiplied by itself, the cross product is zero.

Triple cross product

When a vector is multiplied with the cross product of two other vectors, the result is a triple cross product. Therefore, this result is also a vector quantity. 

Examples of cross products of two vectors

1. When a tap is turned on, the force applied is in two opposite directions and at points diametrically opposite to each other. In the case of a rotating body, torque is the product of the force vector and the radius vector.
2. In a spanner used for tightening or loosening bolts, the handle of the spanner is one vector, and the force applied in a particular direction is the other vector. 

Conclusion

Many areas of engineering make use of cross products of vectors. Knowing how to calculate them and how they affect a system helps predict various outcomes in several physical systems. It is important to remember that we can calculate cross product by multiplying the magnitude of the vectors and the sine of the smaller angle formed by the vectors. Finding out the area of the parallelogram formed by the two vectors is another way to calculate cross product.