Cross product

The cross product of two three-dimensional vectors, also known as the vector product, produces a new vector that is perpendicular to both of the multiplied vectors.

The cross product of two 3D vectors \(\vec{ P }\) and \(\vec{ Q }\), written as \(\overrightarrow{ P \times Q }\), is a vector quantity given by the formula:

\(\overrightarrow{ P \times Q } = \langle P_y Q_z - P_z Q_y, P_z Q_x - P_x Q_z, P_x Q_y - P_y Q_x \rangle \)

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Like the dot product, the cross product has trigonometric significance. Given two 3D vectors \(\vec{ P }\) and \(\vec{ Q }\), the cross product \(\overrightarrow{ P \times Q }\) satisfies the equation:

\( \lVert \overrightarrow{ P \times Q } \rVert = \lVert \vec{ P } \rVert \lVert \vec{ Q } \rVert \sin \alpha \)

where \(\alpha\) is the planar angle between the lines connecting the origin to the points represented by \(\vec{ P }\) and \(\vec{ Q }\), or in simpler terms, it is the angle between vectors \(\vec{ P }\) and \(\vec{ Q }\).

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Any nonzero result of the cross product must be perpendicular to the two vectors being multiplied together, but there are two possible directions that satisfy this requirement. It turns out that the cross product follows a pattern called the right hand rule.

If the fingers of the right hand are aligned with a vector \(\vec{ P }\), and the palm is facing in the direction of a vector \(\vec{ Q }\), then the thumb points along the direction of the cross product \(\overrightarrow{ P \times Q }\).