Further Vectors

Vector product

The vector (cross) product of two vectors is given by:

Where is the angle between and , and is a unit vector perpendicular to and . The direction of is found using the right hand rule, by aligning the first (index) finger with and the second (middle) finger with , the thumb points in the direction of .

From Pure 1, there is also the definition of the cross product in the formula booklet:

Properties

.
If , this means , meaning and are parallel or collinear.
; the vector product is not commutative1.
; the vector product distributes over addition.
; multiplication by a scalar can be 'factored out'.

Applications

Straight lines

The general vector equation of a straight line is , where is a point on the line, is a parameter varied to give different points along the line, and is the direction vector of the line.

Moving terms over gives
Taking the cross product with on both sides gives
Considering gives , another equation for a straight line.

Area of triangles and parallelograms

Consider triangle , which has area (by trigonometry).

Let and . By the definition of the cross product:

Consider parallelogram , which has area (by trigonometry).

Let and . By the definition of the cross product:

1: Anticommutativity of the vector product
The vector product is anticommutative, that is an operation where is the inverse to (in this case the inverse is negating the value).