1. Algebra > Vector Algebra >
History and Terminology > Disciplinary Terminology > Aeronautical Terminology >
Interactive Entries > Interactive Demonstrations >
Vector
A vector is formally defined as an element of a vector space. In the commonly encountered vector space (i.e.,
Euclidean n-space), a vector is given by coordinates and can be specified as . Vectors are
sometimes referred to by the number of coordinates they have, so a 2-dimensional vector is often called a
two-vector, an -dimensional vector is often called an n-vector, and so on.
Vectors can be added together (vector addition), subtracted (vector subtraction) and multiplied by scalars (scalar
multiplication). Vector multiplication is not uniquely defined, but a number of different types of products, such as
thedot product, cross product, and tensor direct product can be defined for pairs of vectors.
A vector from a point to a point is denoted , and a vector may be denoted , or more commonly, . The
point is often called the "tail" of the vector, and is called the vector's "head." A vector with unit length is called
aunit vector and is denoted using a hat, .
When written out componentwise, the notation generally refers to . On the other hand, when
written with a subscript, the notation (or ) generally refers to .
An arbitrary vector may be converted to a unit vector by dividing by its norm (i.e., length; i.e., magnitude),
(1)
giving
(2)
A zero vector, denoted , is a vector of length 0, and thus has all components equal to zero.
Since vectors remain unchanged under translation, it is often convenient to consider the tail as located at the origin
when, for example, defining vector addition and scalar multiplication.
A vector may also be defined as a set of numbers , ..., that transform according to the rule
(3)
where Einstein summation notation has been used,

2. (4)
are constants (corresponding to the direction cosines), with partial derivatives taken with respect to the original and
transformed coordinate axes, and , ..., (Arfken 1985, p. 10). This makes a vector a tensor of tensor
rankone. A vector with components in called an -vector, and a scalar may therefore be thought of as a 1-vector
(or a 0-tensor rank tensor). Vectors are invariant under translation, and they reverse sign upon inversion. Objects that
resemble vectors but do not reverse sign upon inversion are known as pseudovectors. To distinguish vectors
frompseudovectors, the former are sometimes called polar vectors.
A vector is represented in Mathematica as a list of numbers a1, a2, ..., an . Vector addition is then simply written
using a plus sign, e.g., a1, a2, ..., an + b1, b2, ..., bn , and scalar multiplication is indicated by placing a scalar
next to a vector (with or without an optional asterisk), s a1, a2, ..., an .
Let be the unit vector defined in spherical coordinates by
(5)
Then the average value of the -component of the over the surface of the unit sphere is given by
(6)
(7)
(8)
More generally,
(9)
for , , or (indexed as 1, 2, 3), and
(10)
(11)
(12)
Given vectors , , , , the average values of a number of quantities over the unit sphere are given by
(13)
(14)
(15)
(16)
(17)

3. and
(18)
where is the Kronecker delta, is a dot product, and Einstein summation has been used.
A map that assigns each a vector function is called a vector field.
Unit Vector
A unit vector is a vector of length 1, sometimes also called a direction vector (Jeffreys and Jeffreys 1988). The unit
vector having the same direction as a given (nonzero) vector is defined by
where denotes the norm of , is the unit vector in the same direction as the (finite) vector . A unit vector in
the direction is given by
where is the radius vector.
CROSS PRODUCT
The cross product, also called the vector product, is an
operation on two vectors. The cross product of two
vectors produces a third vector which is perpendicular to
the plane in which the first two lie. That is, for the cross
of two vectors, A and B, we place A and B so that their
tails are at a common point. Then, their cross Figure 1 A x B = C
product, A x B, gives a third vector, say C, whose tail is
also at the same point as those of A and B. The
vector C points in a direction perpendicular (or normal)
to both A and B. The direction of C depends on the Right
Hand Rule.

4. If we let the angle between A and B be , then the cross
product of A and B can be expressed as
A x B = A B sin( )
If the components for vectors A and B are known, then
we can express the components of their cross
product, C = Ax B in the following way
Cx = AyBz - AzBy
Cy = AzBx - AxBz Figure 2 B x A = D
Cz = AxBy - AyBx
Further, if you are familiar with determinants, A x B, is
Comparing Figures 1 and 2, we notice that
AxB=-BxA
A very nice simulation which allows you to
investigate the properties of the cross product is
available by clickingHERE. Use the "back" button
to return to this place.

5. Lecture given by Subrata Mukherjee at Cornell University in 1995
Cross Product
We covered the scalar dot product of two vectors in the last lecture and now move on
to the second vector product that can be performed, the Cross Product. The Cross
Product involves taking two vectors and getting as a result another vector which is
perpindicular to both vectors. We use the formula below.
The direction of n follows the right hand rule, which is easy to learn. Tkae the picture
below. Point your thumb in the direction of A, the first vector in the cross product.
Now point your fingers in the direction of B. Your palm will face in the direction of n,
out of the screen.
By this method we can do the Cross Products of i,j, and k.

6. ixj = k
jxk = i
kxi = j
We can also note that AxB = -BxA, since when we switch the position of our thumb
and fingers we have to flip our hand over and our palm then faces in the opposite
direction.
NOTE: MOST RIGHT HANDED PEOPLE TEND TO LOOK AT THEIR LEFT HAND
DURING TESTS, AS THEY'RE BUSY WRITING WITH THE RIGHT HAND. MAKE
SURE TO USE THE RIGHT HAND, BECAUSE THE LEFT HAND WILL GIVE THE
WRONG ANSWER.
Motivations for the Cross Product.
o Mechanics- Torque
To find the torque of a force, you multiply the magnitude of the force
times the magnitude of the torque arm times the sine of the angle
between them. The you define a vector in the direction a right handed
screw would turn if twisted in the direction the force is being applied.
That is exactly the same as the mathematics of the Cross Product.
o Area of a Parrallelogram
Computing the Cross Product
o By Distributive Property

7. Example:
A=i-j
B=i+k
AxB = (i - j) x (i + k) = ixi + ixk - jxi - jxk = (0 + (-j ) - (-k) - i)
= -i - j + k
o By Determinant Formula
Note that both ways return the same answer. (This is very good.)
Applications
1. Find the Area of the shaded
triangle.

8. The area can be found using the fact that the area of the corresponding
parrallelogram is |PAxPB|. So, the area of the triangle is one half of the
area of the paralleogram, or .5*|PAxPB|.
PA = -i + j
PB = -i + j + k
Area = .5*|PAxPB|
= .5*|i + j| = sqrt(2)/2
2. Find a vector perpendicular to triangle PAB.
PAxPB =i+j
3D: Lines, Planes and Segments
0. Lines
v = Ai + Bj + Ck
(x-x0)i + (y-y0)j + (z-z0)k = t(Ai + Bj + Ck)
Taking each component seperately

9. x = x0 + tA
y= y0 + tB
z = z0 + tC
Example:
Find a vector parallel to the line described by the following equations
x=2-t
y= 3 + t
z = 1 - 2t
The Components of the vector are just the coefficients on t, so
v = -i + j - 2k
Let's return to the cube with the triangle inside and find the equation of
the line segment PB.
Base Point: P(1,0,0)
v = PB = -i + j + k
x=1-t
y=t
z=t
Now we must place limits on t, since we have a line segment not a full
line. At P, t=0, and at B(0,1,1) t=1. So 0<=t<=1.
1. Equation of a Pane

10. n is normal to any segment P0P in the plane. Thus the dot product
of n and P0P is 0.
[(x-x0)i + (y-y0)j + (z-z0)k] (dot) (Ai + Bj + Ck) = 0
A(x-x0) + B(y-y0) + C(z-z0) = 0
Ax + By + Cz = Ax0 + By0 + Cz0 = D
Example:
A Plane is described by the equation 2x - 3y = 5
Vector Normal to Plane: 2i - 3j

11. Einstein Cartesian Vector Notation: BAC-CAB Rule
The Einstein notation for Cartesian vectors is a very useful way of dealing
with many complex vector expressions. Mostly one has to recognize and
use a few basic index patterns. This dialogue provides some practice.