
Direction
The direction in which a 2Dvector points can be characterized by a single angle; for 3Dvectors two angles are needed.

Euclidean Space
The name given to all finitedimensional spaces obtained by taking Cartesian products of the real numbers R. They are denoted by R^{n} for n=1,2,3,...

Magnitude
The magnitude of a vector is its length, or distance from the origin.

Projection
The projection of a vector in a particular direction is its "shadow" along that direction. If u is a unit vector, the projection of a vector v in the direction of u is given by a new vector which points in the direction of u and whose magnitude is vƒu: i.e. the projection of v in the direction of u is precisely (vƒu)u.

Righthandrule
This is the standard convention chosen when defining the cross product between two vectors. It states that i×j = k, instead of k, even though both options are equally valid. Once this convention has been chosen, there is no longer any ambiguity about whether the cross product between two vectors points upwards or downwards. (Before this we only knew it had to point in a direction perpendicular to the plane of the original two vectors).

Rotational invariance
A vector quantity (such as the dot product or the cross product) is rotationally invariant if its value remains the same under a rotation of its input vectors. Both the dot product and the cross product are rotationally invariant, while vector addition and scalar multiplication, in general, are not.

Scalar
An ordinary number; whereas vectors have direction and magnitude, scalars have only magnitude. The scalars we will be dealing with will all be real numbers, but other kinds of numbers can also be scalars. 5 miles represents a scalar.

Unit vector
A vector whose length is one. The unit vectors which point in the x, y, and zdirections in typical 3dimensional space are usually denoted by i, j, and k, respectively.

Vector
A twodimensional vector is an ordered pair (a, b) of numbers; a threedimensional vector is an ordered triplet (a, b, c). In other words, points in the plane or in threedimensional space are vectors. These kinds of vectors can also be described as having direction and magnitude: 5 miles to the east represents a vector.

Vector Space
A set that is closed under addition and scalar multiplication. Examples of vector spaces include the Euclidean plane R^{2} and ordinary three dimensional space R^{3}.