Intro
Ever since James Clerk Maxwell discovered the relationship between electricity and magnetism, uniting them under the theory of electromagnetism, vector operations have been crucial in explaining and utilizing the related phenomena.
The cross product is one such operation with applications to, among other things, the generation of power.
As a conducting wire moves perpendicular to a magnetic field, an electric current is induced inside the wire. This is the key concept we make use of in generators that power our electric society, for example in wind turbines.
The same technique is used in electric motors, but in the opposite direction, where electric current is used to produce mechanical motion.
The cross product of the vectors describing the magnetic field and either the electric current or the physical motion gives us the magnitude and direction of the quantity we are after.
Concept
As opposed to the dot product, which is another method of multiplication defined for vectors of any size, the cross product only works for vectors in three dimensions.
Intuitively, this makes sense because the cross product between any two vectors is always perpendicular to both of them. Only in three dimensions is there a unique perpendicular direction.
The cross product is a way of multiplying two vectors, and the result is a third vector perpendicular to both
The only thing we might not know about the direction of the cross product beforehand is whether it is positive or negative. This is determined by the magnitude and its sign, which we find by carrying out the calculation.
Geometrically, we can interpret the magnitude of the cross product of two vectors as the area of a parallelogram having the vectors as its sides.
Math
We can use the following formula to carry out the cross product, element-by-element:
Alternatively, the cross product can be derived using the magnitudes of the vectors, and , and the angle between them:
where denotes a unit vector in the direction perpendicular to and .
Cross product
Multiplication between two vectors is not defined, but there are two definitions where multiplication is still used between the elements; dot product (or scalar product) and cross product (or vector product). The dot product is a simple product sum, while the cross product derives from the definition of the determinant, which is recognizable. While the dot product of two vectors results in a scalar, the cross product between two vectors is a new vector that is orthogonal to the two vectors. One difference is that the cross product is only defined for three dimensions (), while the dot product is defined for all dimensions of space.
We summarize:
Let and be vectors in . Then the cross product of and is noted as and is defined as:
or equivalently:
The evolution of the cross product can be deduced from the definition of the -determinant. Therefore, let the cross product be expressed as the linear combination of the unit vectors , and :
where , and are cofactors to the -determinant as follows:
The following properties apply to the cross product:
Let , and be vectors in and be a scalar. Then we have:
Area
The formula for the area of a parallelogram is the base multiplied by the height. In fact, the cross product of and is related to the surface that the two vectors span, namely that its result is equal to the area of its surface. We have the statement:
Let and be non-zero vectors in and let be the angle between these two. Let be the area that the two span. Then it applies that:
First we need to prove that:
The right side follows the formula "the base multiplied by the height". The term yields the height via basic trigometry, with the base being . From the Pythagorean trigonometric identity we have:
Okay, here it comes:
Triple product
As an extension of how the cross product is related to the surface spanned by its two vectors, we define the triple product as follows; Let , and be three non-zero vectors in . Then the triple product is defined as:
For the geometric interpretation, corresponds to the volume of the parallelepiped that the three vectors span.
We have the general form of the volume , that is the base is multiplied by the height , where the base is the surface that is stretched by the vectors and .
Thus, the geometric interpretation is derived.
Normal vectors
A normal is a vector whose direction is orthogonal (forms a right angle) to another object. This object can be another vector, a plane, a hyperplane or even a geometric object such as a non-linear surface. The latter is not treated in a course of linear algebra, but is an obvious element of multivariable analysis.
The easiest way to create a normal vector for a plane is to take two vectors belonging to the plane, and , and calculate their cross product:
