Cross (Vector) Product: The cross product, also known as the vector product or tensor product, of two vectors is a new vector that represents a linear trans...
Cross (Vector) Product:
The cross product, also known as the vector product or tensor product, of two vectors is a new vector that represents a linear transformation on the plane of the original vectors.
Definition:
The cross product of two vectors (\overrightarrow{a}) and (\overrightarrow{b}) is a new vector (\overrightarrow{a}\times\overrightarrow{b}) that is perpendicular to both (\overrightarrow{a}) and (\overrightarrow{b}) and has the same direction as the normal to the plane of the vectors.
Formula:
Geometric Interpretation:
The cross product of two vectors (\overrightarrow{a}) and (\overrightarrow{b}) can be thought of as the projection of (\overrightarrow{a}) onto the subspace of (\overrightarrow{b}) that is perpendicular to (\overrightarrow{a}).
Properties:
The cross product is linear, meaning that (\overrightarrow{a}\times(\overrightarrow{b} + \overrightarrow{c}) = \overrightarrow{a}\times\overrightarrow{b} + \overrightarrow{a}\times\overrightarrow{c}).
The cross product is always perpendicular to both (\overrightarrow{a}) and (\overrightarrow{b}).
The magnitude of the cross product is equal to the product of the magnitudes of the two original vectors, and it has the same direction as the cross product of two vectors if they are perpendicular.
Examples:
(\overrightarrow{a} = (1, 2, 3)) and (\overrightarrow{b} = (4, 5, 6)) then (\overrightarrow{a}\times\overrightarrow{b} = (-6, 12, -15)).
(\overrightarrow{a} = (1, 0, 0)) and (\overrightarrow{b} = (0, 1, 0)) then (\overrightarrow{a}\times\overrightarrow{b} = (0, 0, 1)).
(\overrightarrow{a} = (1, 2, 3)) and (\overrightarrow{b} = (4, 5, 6)) then (\overrightarrow{a}\times\overrightarrow{b} = (6, -30, 30))