Vector
The word vector has many different definitions, depending on who is defining it and in what context. Physicists will often refer to a vector as "a quantity with a direction and magnitude." For Euclidean geometers, a vector is essentially a directed line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space.
A vector is usually graphically represented as an arrow. Vectors can be uniquely described in many ways. The two most common is (for 2-dimensional vectors) by describing it with its length (or magnitude) and the angle it makes with some fixed line (usually the x-axis) or by describing it as an arrow beginning at the origin and ending at the point . An
-dimensional vector can be described in this coordinate form as an ordered
-tuple of numbers within angle brackets or parentheses,
. The set of vectors over a field is called a vector space.
Contents
Description
Every vector has a starting point
and an endpoint
. Since the only thing that distinguishes one vector from another is its magnitude or length, and direction, vectors can be freely translated about a plane without changing. Hence, it is convenient to consider a vector as originating from the origin. This way, two vectors can be compared by only looking at their endpoints. This is why we only require
values for an
dimensional vector written in the form
. The magnitude of a vector, denoted
, is found simply by
using the distance formula.
Addition of Vectors
For vectors and
, with angle
formed by them,
.
|
Addition of vectors |
From this it is simple to derive that for a real number ,
is the vector
with magnitude multiplied by
. Negative
corresponds to opposite directions.
Properties of Vectors
Since a vector space is defined over a field , it is logically inherent that vectors have the same properties as those elements in a field.
For any vectors ,
,
, and real numbers
,
(Commutative in +)
(Associative in +)
- There exists the zero vector
such that
(Additive identity)
- For each
, there is a vector
such that
(Additive inverse)
(Unit scalar identity)
(Associative in scalar)
(Distributive on vectors)
(Distributive on scalars)
Vector Operations
Dot (Scalar) Product
Consider two vectors and
in
. The dot product is defined as
, where
is the angle formed by the two vectors. This also yields the geometric interpretation of the dot product: from basic right triangle trigonometry, it follows that the dot product is equal to the length of the projection (i.e. the distance from the origin to the foot of the head of
to
) of
onto
times the length of
. Note that the dot product is
if and only if the two vectors are perpendicular.
Cross (Vector) Product
The cross product between two vectors and
in
is defined as the vector whose length is equal to the area of the parallelogram spanned by
and
and whose direction is in accordance with the right-hand rule. Because of this,
, where
is the angle formed by the two vectors, and from the right-hand rule condition,
. Also,
gives that
.
If and
, then the cross product of
and
is given by
![$\bold{a}\times\bold{b}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}.$](http://latex.artofproblemsolving.com/b/e/b/bebf3df76dfd91f84b9cf2e0158e42dc7f2cbc7e.png)
where are unit vectors along the coordinate axes, or equivalently,
. Also,
Triple Scalar Product
The triple scalar product of three vectors is defined as
. Geometrically, the triple scalar product gives the signed volume of the parallelepiped determined by
and
. It follows that
![$(\bold{a}\times\bold{b})\cdot \bold{c} = (\bold{c}\times\bold{a})\cdot \bold{b} = (\bold{b}\times\bold{c})\cdot \bold{a}.$](http://latex.artofproblemsolving.com/2/f/1/2f1162b7eb4c301897dba471b3f4372b0a70cec7.png)
It can also be shown that
![$(\bold{a}\times\bold{b})\cdot \bold{c} = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}.$](http://latex.artofproblemsolving.com/f/a/6/fa6e7cbf53bb81ffed2d33a8e63a2942274b9def.png)
Triple Vector Product
The vector triple product of is defined as the cross product of one vector, so that
, which can be remembered by the mnemonic "BAC-CAB" (this relationship between the cross product and dot product is called the triple product expansion, or Lagrange's formula).
Bivectors
Sometimes when working with vectors (especially the cross-product) it can be hard to understand what is happening. For example, imagine describing a wheel on a car rotating with a vector. The resulting vector does not describe the rotation happening intuitively. This is where bivectors can help.
Just like vectors are oriented lines, bivectors are oriented areas. Consider two vectors and
. The bivector formed by these vectors (denoted
) represents the parallelogram formed by
and
. Thus, the magnitude
represents the area formed by this parallelogram.
Calculating the bivector requires only three properties. First, the bivector since there is no way to form a parallelogram. Second the bivector
. Finally, the wedge product (
operation) is distributed.
For 2D vectors and
, the resulting bivector is
\begin{align}
u \wedge v &= (u_1 e_1 + u_2 e_2)\wedge (v_1 e_1 + v_2 e_2) \\
&= u_1 e_1\wedge (v_1 e_1 + v_2 e_2) + u_2 e_2\wedge (v_1 e_1 + v_2 e_2) \\
&= u_1 e_1\wedge v_1 e_1 + u_1 e_1\wedge v_2 e_2 + u_2 e_2\wedge v_1 e_1 + u_2 e_2\wedge v_2 e_2 \\
&= u_1 v_2 e_1\wedge e_2 - u_2 v_1 e_1\wedge e_2 \\
&= (u_1 v_2 - u_2 v_1)e_1\wedge e_2 \\
\end{align}
where
represents the bivector for a unit square. This result is the same as calculating the determinate! Thus bivectors can help to calculate determinates.
Now consider 3D vectors and
. The resulting bivector becomes
\begin{align}
u \wedge v &= (u_1 e_1 + u_2 e_2 + u_3 e_3)\wedge (v_1 e_1 + v_2 e_2 + u_3 e_3) \\
&= u_1 e_1\wedge (v_1 e_1 + v_2 e_2 + v_3 e_3) + u_2 e_2\wedge (v_1 e_1 + v_2 e_2 + v_3 e_3) +u_3 e_3\wedge (v_1 e_1 + v_2 e_2 + v_3 e_3) \\
&= u_1 e_1\wedge v_1 e_1 + u_1 e_1\wedge v_2 e_2 + u_1 e_1\wedge v_3 e_3 + \\
& \: \: \: u_2 e_2\wedge v_1 e_1 + u_2 e_2\wedge v_2 e_2 + u_2 e_2\wedge v_3 e_3 + \\
& \: \: \: u_3 e_3\wedge v_1 e_1 + u_3 e_3\wedge v_2 e_2 + u_3 e_3\wedge v_3 e_3 \\
&= u_1 e_1\wedge v_2 e_2 + u_1 e_1\wedge v_3 e_3 + u_2 e_2\wedge v_1 e_1 + u_2 e_2\wedge v_3 e_3 + u_3 e_3\wedge v_1 e_1 + u_3 e_3\wedge v_2 e_2 \\
&= u_1v_2 e_1\wedge e_2 - u_1 v_3 e_3\wedge e_1 - u_2 v_1 e_1\wedge e_2 + u_2 v_3 e_2\wedge e_3 + u_3v_1 e_3\wedge e_1 - u_3v_2 e_2\wedge e_3 \\
&=(u_2 v_3 - u_3v_2) e_2\wedge e_3 +(u_3 v_1 e_3 - u_1 v_3) e_3\wedge e_1 + (u_1v_2 - u_2 v_1) e_1\wedge e_2 \\
\end{align}
which is the same as the cross product
! Thus, instead of using determinates to calculate the cross-product as above, it is possible to use bivectors instead.
Exercise
Through the use of the bivector, it is possible to extend the definition of the cross-product to higher dimensions. Try to calculate the 4D cross-product using the bivector properties mentioned above.