Vector Dot Products and Cross-Products

Previous material introduced you to vectors and some manipulation of them in the forms of vector addition and subtraction. This material will focus on the product of two vectors. Specifically we will look at the Vector Dot Product and the Vector Cross Product. We will use a small bold letter with a line over the top to represent a vector. Scalars will not be in bold and will not have the line over the top.

Dot Products

The dot product of two vectors u = (u1, u2) and v = (v1, v2) is defined as u • v= u1*v1 + u2*v2. I have used vectors with two components (x, y) but what is said here is also extendable to vectors with three components (x, y and z). If we calculate the dot product of two vectors that are perpendicular to one another the dot product will be zero. When we take the dot product of two vectors the result is a scalar. For dot products the order does not matter. u • v = v • u. Here is a sample dot product. The tool located below can be used to calculate results using other vectors.
Example - Dot Product
r = (2, -3)
s = (4 ,8)
r • s = 2*4 + (-3)*8 = 8 - 24 = -16

Cross Products

The cross product of two vectors u = (u1, u2, u3) and v=(v1, v2, v3) is defined as u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1). When we take the cross product to two vectors the result is a new vector that is perpendicular to the two vectors we started with. You can test that the new vector is perpendicular to the starting vectors by calculating its dot product with them. The dot product of a starting vector with the resulting vector should be zero. You might also ask yourself if the order of taking the cross product matters. For cross products the order does matter. The direction of the vector will reverse if we change the order of taking the cross products. Here is an example that illustrate the cross product. The tool located below will calculate cross product for (x, y) and (x, y, z) vectors. You might try the example in the tool as it is shown and then reverse the order and see if that changes your answers.
Example - Cross Product
r = (3, -1, 2)
s = (4, -5, 6)
rxs= (-1*6- 2*-5, 2*4 - 3*6, 3*-5 -1*4) = (4, -10, -11)

Tools for Calculating Vector Dot and Cross Products

Input values for first vector as x,y,z
Input values for second vector as x,y,z
Click to compute dot and corss products and angle
Dot product is displayed in field on the right
The angle between the vectors in degrees and radians is:
The vector cross product is given as (x, y, z)