Skip Ahead:

Main Vector Page | x-axis Vector Cross Product | y-axis Vector Cross Product | 2D Vector Cross Product | 3D Vector Cross Product

Vector Cross Product

Vector Cross Product in 3D

The Cross Product of two vectors is one of two methods of vector multiplication, the other is the dot product. A cross product is denoted by the multiplication sign (x) between two vectors. The cross product of two vectors is a third vector that not only is perpendicular to the two original vectors, but its magnitude (length) is given by the area of the parallelogram between them and its direction can be determined by the right-hand thumb rule. It is worth noting that the cross product of two vectors is also known as a vector product as the resultant of the cross product of vectors is a vector quantity.

Below is the same simulation as the one previously of the cross product in 3D. There is a red vector with a x-axis component vector of 1, a y-axis component vector of -1, and a z-axis component vector of 2. There is then a blue vector, with a x-axis component vector of -2, a y-axis component vector of 1, and a z-axis component vector of 3. Since these vectors are on the x, y, and z axis the cross of these component vectors will result in a vector on all of these axis towards the screen, since these component vectors are both positive and negative. This new vector, which is purple, is the cross of the red and blue vectors. As a student you will be able to change this vector's angle, components, direction and magnitude to alter the result of the vector's cross product.

Red Vector:

x, y, z:

Blue Vector:

x, y, z:

Cross Product:

The vectors in the simulation can be represented in numerous ways. Here are the three most common ways of portraying them:

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 2 \end{bmatrix} \]

or

\[ \vec{A} = 1\hat{i} - 1\hat{j} + 2\hat{k} \]

or

\[ \vec{A} = \langle 1, -1, 2 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ -2 & 1 & 3 \end{bmatrix} \]

or

\[ \vec{B} = -2\hat{i} + 1\hat{j} + 3\hat{k} \]

or

\[ \vec{B} = \langle -2, 1, 3 \rangle \]

The cross product can be calculated using the following equations:

\[ \vec{A} \times \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{bmatrix} \]

or

\[ \vec{A} \times \vec{B} = (A_2 B_3 - A_3 B_2)\hat{i} + (A_1 B_3 - A_3 B_1)\hat{j} + (A_1 B_2 - A_2 B_1)\hat{k} \]

Now let's see what happens when we calculate the cross product of the red vector and blue vector:

\[ \vec{A} = 1\hat{i} - 1\hat{j} + 2\hat{k} \]

\[ \vec{B} = -2\hat{i} + 1\hat{j} + 3\hat{k} \]

\[ \vec{A} \times \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 2 \\ -2 & 1 & 3 \end{bmatrix} \]

\[ \vec{A} \times \vec{B} = ((-1)(3) - (2)(1))\hat{i} + ((2)(-2) - (1)(3))\hat{j} + ((1)(1) - (-1)(-2))\hat{k} \]

\[ \vec{A} \times \vec{B} = (-3 - 2)\hat{i} + (-4 - 3)\hat{j} + (1 - 2)\hat{k} \]

\[ \vec{A} \times \vec{B} = -5\hat{i} - 7\hat{j} - 1\hat{k} \]

\[ |\vec{A} \times \vec{B}| = \sqrt{75} \]

As we can see, the result of the cross product is √75, just like in the simulation.

You have now learned about the cross product. Go back now to the vector page to review any material that still may be confusing.

Go Back to 2D Vector Cross Product Go Back to Vector Page