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Vector Cross Product

Vector Cross Product in 1D y-axis

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 a simulation of the cross product in the y-axis. There is a red vector with a variable y-axis component vector based on the movement of your mouse, and a second vector, a blue vector with a y-axis component vector of 12. When vectors are on the same axis, and the cross product is being used, the result is always 0. This is because the cross of two of the same component vectors always equals 0. Also, visually there is no parallelogram that can be made by two vectors that are on the same axis.

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

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 23.5 & 0 \end{bmatrix} \]

or

\[ \vec{A} = 0\hat{i} + 23.5\hat{j} + 0\hat{k} \]

or

\[ \vec{A} = \langle 0, 23.5, 0 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 12 & 0 \end{bmatrix} \]

or

\[ \vec{B} = 0\hat{i} + 12\hat{j} + 0\hat{k} \]

or

\[ \vec{B} = \langle 0, 12, 0 \rangle \]

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:

\[ \vec{A} = 0\hat{i} + 23.5\hat{j} + 0\hat{k} \]

\[ \vec{B} = 0\hat{i} + 12\hat{j} + 0\hat{k} \]

\[ \vec{A} \times \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 23.5 & 0 \\ 0 & 12 & 0 \end{bmatrix} \]

\[ \vec{A} \times \vec{B} = ((23.5)(0) - (0)(12))\hat{i} + ((0)(0) - (0)(0))\hat{j} + ((0)(12) - (23.5)(0))\hat{k} \]

\[ \vec{A} \times \vec{B} = (0 - 0)\hat{i} + (0 - 0)\hat{j} + (0 - 0)\hat{k} \]

\[ \vec{A} \times \vec{B} = 0\hat{i} + 0\hat{j} + 0\hat{k} \]

\[ \vec{A} \times \vec{B} = 0 \]

As we can see, the result of the cross product is 0, just like in the simulation.

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