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Vector Cross Product

Vector Cross Product in 2D

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 2D. There is a red vector and a blue vector with variable x and y axis component vectors based on the movement of the slider in the simulation. Since these vectors are on the x and y axis, the cross of these component vectors will result in a vector solely on the z-axis towards the screen, since these component vectors are positive. As you increase the x and y components of the two vectors, the z-axis purple vector will become longer and more visible at the origin of the screen, (0, 0). As you decrease the x and y components of the two vectors, the z-axis purple vector will become shorter and less visible at the origin of the screen (0, 0). As a result of the cross product of these two vectors, you can also see a coral parallelogram, which also varies in area as you increase or decrease the component vectors of the red and blue vectors. The area of this parallelogram is equal to the length of the purple vector.

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

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 14.4 & 3.6 & 0 \end{bmatrix} \]

or

\[ \vec{A} = 14.4\hat{i} + 3.6\hat{j} + 0\hat{k} \]

or

\[ \vec{A} = \langle 14.4, 3.6, 0 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 5.4 & 7.2 & 0 \end{bmatrix} \]

or

\[ \vec{B} = 5.4\hat{i} + 7.2\hat{j} + 0\hat{k} \]

or

\[ \vec{B} = \langle 5.4, 7.2, 0 \rangle \]

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:

\[ \vec{A} = 14.4\hat{i} + 3.6\hat{j} + 0\hat{k} \]

\[ \vec{B} = 5.4\hat{i} + 7.2\hat{j} + 0\hat{k} \]

\[ \vec{A} \times \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 14.4 & 3.6 & 0 \\ 5.4 & 7.2 & 0 \end{bmatrix} \]

\[ \vec{A} \times \vec{B} = ((3.6)(0) - (0)(7.2))\hat{i} + ((14.4)(0) - (0)(5.4))\hat{j} + ((14.4)(7.2) - (3.6)(5.4))\hat{k} \]

\[ \vec{A} \times \vec{B} = (0 - 0)\hat{i} + (0 - 0)\hat{j} + (103.68 - 19.44)\hat{k} \]

\[ \vec{A} \times \vec{B} = 0\hat{i} + 0\hat{j} + 84.24\hat{k} \]

\[ \vec{A} \times \vec{B} = 84.24 \]

As we can see, the result of the cross product is 84.24, just like in the simulation.

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