4.2. INTRODUCTION TO CROSS PRODUCT

The method of Cross Product is not as simple as the Dot Product but it also has specific functionalities. The Cross Product of two vectors result in another vector - with both magnitude and direction.

ALGEBRAIC DEFINITION

The algebraic definition of the Cross Product utilizes the determinant of a matrix. The procedure of the method is somehow similar with the co-factor procedure in advance math topics but it is a little bit different in some parts.

Consider vectors A and B with i, j, and k axes, the matrix representation of the cross product will turn like below:

                                        


                              

With the simplified matrix, take the determinants of each axis to have the numerical value which is the cross product.

PROPERTIES

1. Cross product is non-commutative: 
2. It is distributive:                            
3. The cross product of a vector on its own = zero: 

GEOMETRIC DEFINITION

1. While Dot Product measures the similar vectors, Cross Product measures the difference of two vectors. It measures orthogonality. ThusCross Product acts as a dual for Dot Product.

                                                

where:    is the scalar part of the cross product
                                         is the direction of the cross product

ORTHOGONALITY

From the geometric definition, the Cross Product of two vectors is a vector which is orthogonal to both vectors, that is, it is perpendicular to both vectors. And with the property which is non commutative, the direction of vectors should be observed.

APPLICATIONS

1. The cross product can be used to determine the shortest distance in a system.
2. By using Scalar Triple Product, the common perpendicular of two vectors can be determined.

Example 4.2.1: Cross Product: Shortest Distance

Example 4.2.2: Cross Product: Shortest Distance

Example 3: Common Perpendicular

Example 4: Common Perpendicular

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