Cross product (Vector Product)

The cross product (or vector product) is a mathematical operation that produces a vector that is perpendicular to the plane of the two input vectors.

It is defined by the following formula: $\vec{A}\times \vec{B}=|A|\ast |B|\ast sin(\theta )\ast n$

• where $A$ and $B$ are input vectors.
• $|A|$ and $|B|$ are the magnitudes of the vectors.
• $\theta$ is the angle between the vectors.
• $n$ is a unit vector perpendicular to the plane of the input vector often known as the normal vector.

Example

Suppose you have two vectors, A and B, with the following components: $A=(3,4)$ $B=(2,1)$

First, we need to calculate the magnitudes of the vectors:

$|A|=\sqrt{3^2+4^2}=\sqrt{25}=5$ $|B|=\sqrt{2^2+1^2}=\sqrt{5}=2.236$

Next, we need to calculate the angle between the vectors:

$\theta =\arccos \left(\frac{A\cdot B}{|A|\ast |B|}\right)$ $=\arccos \left(\frac{3_2+4_1}{5\ast 2.236}\right)$ $=\arccos \left(\frac{7}{11.18}\right)$ $=57.22$ degrees

Finally, we need to calculate the normal vector, n:

$n=\frac{A\times B}{|A\times B|}$

To calculate $A\times B$, we use the following formula:

$A\times B=(A1\ast B2-A2\ast B1,A2\ast B0-A0\ast B2,A0\ast B1-A1\ast B0)$ $=(3\ast 1-4\ast 2,4\ast 0-3\ast 0,3\ast 0-4\ast 0)$ $=(-5,0,0)$

Then we can calculate the magnitude of $A\times B$:

$|A\times B|=\sqrt{(-5{)}^2+0^2+0^2}=\sqrt{25}=5$

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