Dot Product, a·b = |a| |b| cos θ, where θ is the angle between a and b.

Cross Product is a vector product a×b, which is a vector normal (i.e. perpendicular) to the plane containing a and b, whose magnitude is |a×b| = |a| |b| (sin θ), where θ is the angle between a and b.

Triple Product a·(b×c) is a scalar representing the "signed volume" of a parallelepiped

Vector Product Identities and Triple Cross Identity give various identities involving the rearrangement of vectors in dot- and cross-products

See also these related pages for methods of solving problems using vectors.

Interpretations of a Vector

A vector is an abstract mathematical concept that can be interpreted in a number of ways.  In two dimensions, a vector is often written as (a,b) where a,b are real numbers.  For example, (3,4) is a vector.  In three dimensions, a vector can be written (a,b,c), and you can see the concept extends easily to larger numbers of dimensions.

A geometrical interpretation of a vector is a ray that has a length (or magnitude) and direction, but no fixed starting point.  The "modulus" of the vector is its length, and it can be calculated using the distance formula -- the square root of the sum of the squares of the elements of the vector.  The direction of a vector is often given by naming a "unit vector" (a vector whose length is 1) that points in the same direction as the given vector.

Vector arithmetic

The most common arithmetic operations that are performed using vectors are

scalar product -- multiplying every element of the vector by a given scalar

Dot Product, a·b = |a| |b| cos θ, where θ is the angle between a and b.

Cross Product is a vector product a×b, which is a vector normal (i.e. perpendicular) to the plane containing a and b, whose magnitude is |a×b| = |a| |b| (sin θ), where θ is the angle between a and b.

Point - vector duality

In geometry, it is often convenient to use vector arithmetic to represent points.  A vector, by its definition, has no fixed starting point, but if we imagine the starting point of a vector to be the origin, then the endpoint of the vector represents a particular point.  In this manner, every vector can be said to identify a unique point, which is the endpoint of the vector when its starting point is the origin.

Related pages in this website

Matrix Math

Geometry and Trigonometry, and in particular, the Points and Lines section.

Triangle Area Using Vectors, part 1

Triangle Area using Vectors, part 2 -- proof that endpoints of vectors u, v, and w form a triangle whose area is (1/2)(u×v + v×w + w×u).

The webmaster and author of the Math Help site is Graeme McRae.
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