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 commutative^[1].
; 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:

Scalar triple product

The scalar triple product of three vectors is given by:

Note how swapping the order of the vectors in a circular order does not affect the scalar triple product. However, swapping any two vectors, e.g. would give negate the value (by the anticommutativity of the cross product). The scalar triple product can also be defined by the determinant of a matrix:

Applications

The volume of a parallelepiped with sides , . and is given by the magnitude of the scalar triple product: .
The volume of a tetrahedron with sides , . and is given by one-sixth of the magnitude of the scalar triple product : .
If the scalar triple product is zero, then the three vectors , . and are coplanar, as they form a tetrahedron with zero volume (so they lie along a plane).

we also ran out of budget for tikz diagrams here :(
no seriously do you know how hard 3d is

Footnote on 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).↩︎