The vector (or cross) product of the vectors a and b is defined to be:

a \times b = |a| |b| \sin \theta \hat{n}

where \theta is the angle between the vectors and \hat{n} is a unit vector perpendicular to both a and b.

The direction of \hat{n} is given by the right-hand rule - see the diagram.

The magnitude the cross product is equal to the area of the parallelogram that the vectors span.

If i, j and k are the familiar unit vectors, then:

i \times j = k, j \times k = i, k \times i = j and

i \times i = 0, j \times j = 0, k \times k = 0

If a = a_1i+a_2j+a_3k \quad and b = b_1i+b_2k+b_3k \quad then: a \times b = (a_2b_3-a_3b_2)i + (a_3b_1-a_1b_3)j + (a_1b_2-a_2b_1)k

a \times b = |a| |b| \sin \theta \hat{n}

where \theta is the angle between the vectors and \hat{n} is a unit vector perpendicular to both a and b.

The direction of \hat{n} is given by the right-hand rule - see the diagram.

The magnitude the cross product is equal to the area of the parallelogram that the vectors span.

If i, j and k are the familiar unit vectors, then:

i \times j = k, j \times k = i, k \times i = j and

i \times i = 0, j \times j = 0, k \times k = 0

If a = a_1i+a_2j+a_3k \quad and b = b_1i+b_2k+b_3k \quad then: a \times b = (a_2b_3-a_3b_2)i + (a_3b_1-a_1b_3)j + (a_1b_2-a_2b_1)k

## Software/Applets used on this page

## Glossary

### cross product

For vectors a and b, the cross product is the vector c whose magnitude is ab sin C, where C is the angle between the directions of the vectors, and which is perpendicular to both a and b.

### magnitude

A measure of the size of a mathematical object

### perpendicular

one line being at right angles to another.

### rule

A method for connecting one value with another.

### union

The union of two sets A and B is the set containing all the elements of A and B.

### unit vector

A vector with magnitude equal to 1.

### vector

A mathematical object with magnitude and direction.