Review of Vectors
Terms
Direction
-
The direction in which a 2D-vector points can be characterized by a single
angle; for 3D-vectors two angles are needed.
Euclidean Space
-
The name given to all finite-dimensional 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
.
Right-hand-rule
-
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
z
-directions in typical 3-dimensional space are usually denoted by
i
,
j
, and
k
, respectively.
Vector
-
A two-dimensional vector is an ordered pair
(a, b)
of numbers; a
three-dimensional vector is an ordered triplet
(a, b, c)
. In other words,
points in the plane or in three-dimensional 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
.
