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### Contents of this section:  Dot Product Cross Product Triple Product Vector Product Identities Triple Cross Identity

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.

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### 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 this Math Help site is Graeme McRae.