Cross product
The Cross Product
The cross product is a vector which is more complex than the dot product. To find the vector,
use a three-by-three determinant.
● The first row of the determinant is the unit vectors i, j, and k.
● The second row is the components of b, so b1, b2j, and b3k.
● The cross product satisfies certain properties similar to the dot product.
If we look at a cross b and commute a and b to b, the determinant changes sign. For example, a
cross a is equal to minus b cross b.
The length of a cross b is the magnitude of a times the length of a vector. The magnitude is
a1b2 sin(θ), which is a coordinate independent expression.
To determine the direction of a cross b, use the right-hand rule. Put your fingers in the direction
of a and your thumb will point in the direction of b towards you.
If a is parallel to b, then θ equals zero. If a is perpendicular to b, then a cross b equals zero. The
magnitude of the cross product of a cross b is the length and direction of the vector.
Jeff Chasnov calls the cross product the vector product. The cross product is often called the
vector product or product of two dimensions.
The Cross Product
The cross product is a vector which is more complex than the dot product. To find the vector,
use a three-by-three determinant.
● The first row of the determinant is the unit vectors i, j, and k.
● The second row is the components of b, so b1, b2j, and b3k.
● The cross product satisfies certain properties similar to the dot product.
If we look at a cross b and commute a and b to b, the determinant changes sign. For example, a
cross a is equal to minus b cross b.
The length of a cross b is the magnitude of a times the length of a vector. The magnitude is
a1b2 sin(θ), which is a coordinate independent expression.
To determine the direction of a cross b, use the right-hand rule. Put your fingers in the direction
of a and your thumb will point in the direction of b towards you.
If a is parallel to b, then θ equals zero. If a is perpendicular to b, then a cross b equals zero. The
magnitude of the cross product of a cross b is the length and direction of the vector.
Jeff Chasnov calls the cross product the vector product. The cross product is often called the
vector product or product of two dimensions.
