
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 v·u: i.e. the projection of v in the
direction of u is precisely (v·u)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}.