Vectors--background

It was realized in late 19th and early 20th century that vectors are the ideal tools for the exposition and simplification of many important ideas in geometry and physics.

There are three different ways to introduce vector algebra: geometically, analytically and axiomatically.

 

Geometric Representation

In the geometric presentation, vectors are represented by directed line segments, or arrows. Algebraic operations or vectors, such as addition, subtraction, and multiplication by real numbers, are defined and studied by geometric methods.

 

Analytic Representation

In the analytic presentation, vectors and their operations are described in terms of numbers, or components. The analytic description of vectors is a natural outgrowth of the geometric description of vectors is a natural outgrowth of the geometric description coupled with a coordinate system.

 

Axiomatic Representation

The axiomatic presentation does not involve algebra or geometry, but rather involve a set of undefined concepts called axioms which the vectors and vector operations satisfy.

 

 

Examples

B=(b1,b2)

A=(a1,a2)

 

 

 

 

Vector AB: A is the initial poin, B is the terminal point . The vector has coordinates <b1-a1, a2-b2>. The order of the subtraction is important; note that vector BA has B as initial point, A as terminal point, and coordinates <a1-b1, a2-b2>.

             

                 B=(b1,b2)

A=(a1,a2)

 

Any vector that has the origin as its initial point is represented by the coordinated of its terminal point.

 

O=(0,0), C=(2,3), D=(2,-1)

OC = <2-0,3-0> = <2,3>

CO = <0-2,0-3> = <-2,-3> = -OC

OD = <2-0,-1-0> = <2,-1> = -OD

 

 

Two vectors AB, CD are equivalent if (*) B - A = C- D. Such vectors will be parallel, have equal length, and point in the same direction. Equation (*) can be rewritten as A+D=B+C, which tells us that the opposite vertices of a parallelogram have the same sum.

 

 

D

C

B

A

 

 

 

D= B+C

C

B

O

 

 

 

 

 

 

 

 

 

 

If we have the origin O as one of the points, say, point A, then D = B + C.

This is the parallelogram law, i.e., the vector from the origin to D = B + C is a diagonal of

the parallelogram determined by OB and OC.

Vector subtraction is illustrated as follows:

Note that both vectors B - A have the same length and direction.

Mutiplication of a vector by a scalar has the following interpretation:

if B = cA, B has length |c| times the length of A; it points in the same direction as A if

c is positive, and in the opposite direction if c is negative.

Exercises

Let A = <2,1>, B = <1,3> and C = xA + yB, where x and y are scalars.

Draw the geometric vector from the origin to C for the following values of x and y:

1) Solution:

 

 

 

 

 

 

 

 

 

 

2)

3)

4) x = 2, y = -1

 

 

 

The Dot Product

We now introduce a different kind of multiplication that can be performed on vectors of length n:

Definition: If are two vectors, then their dot product is denoted by and is defined by the equation .

Thus, to compute the dot product of A and B, we multiply the corresponding components of A and B and then add all the products. This multiplication has the following algebraic properties:

Theorem: For all vectors A,B,C, and all scalars k, we have the following properties:

  1. (commutative)
  2. (distributive)
  3. (homogeneity)
  4. (positivity)

 

Ex/Let A = <1,2>, B = <-1,2>, C = <0,1> be three vectors.

 

Given vectors A = <-2,2>, B = <-1,0>, C = <3,1> , compute

 

 

 

The Length/Norm of a Vector

The vector from the origin to a point has length .

 

 

 

b

a

 

 

 

 

 

 

We symbolize this as follows:

The properties of the dot product lead to corresponding properties of the norms:

 

 

The triangle inequality takes on the following form:

If A and B are vectors, .

Geometrically, this has the following interpretation:

 

A+B

B

A

 

 

 

 

Ex/Let A = <1,2>, B = <-1,2>, C = <0,1> be three vectors.

For A and B, the triangle inequality is as follows:

and the Triangle inequality is verified

 

 

 

 

Given vectors A = <-2,-2>, B = <3,10>, C = <0,-5> ,

compute , and verify the triangle inequality for B and C.

Orthogonality and Projections

Definition: Two vectors are orthogonal, or perpendicular (i.e., they meet in a right angle), if .

Note that

Thus, if two vectors are orthogonal, we have .

 

 

B A+B

A

Ex/ Find all vectors orthogonal to A = <1,2> having the same length as A.

, so let B = <c , d> . We know

Also, . Substituting c = -2d into the first equation,

We obtain . If d =1, c=-2, so one orthogonal vector is <-2,1>. If d = -1, then c=2, so

<2,-1>.

 

 

Find all vectors orthogonal to <1,-2> having the same length.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The dot product of two vectors has an interesting geometric interpretation:

Consider two vectors A and B which make an angle with each other.

 

 

A

 

B

 

 

 

 

A = tB + C

C

 

tB = projection of A along B

Note that tB is a scalar multiple of B is called the projection of A along B. In the drawing, t is positive, because , and C is drawn perpendicular to B.

We can use dot products to express t in terms of A and B. First, write A = tB + C and then take the dot product of each member with B to obtain . However, because C is perpendicular to B, which leads to . Further, it can be shown that the dot product of two nonzero vectors is the product of three numbers: the length of each vector and the cosine of the angle between them, i.e., (from figure) .

 

 

 

 

 

 

 

Ex/ Determine if the vectors are orthogonal:

<4,0>, <1,1>

(4)(1) + (0)(1) = 4 0.

Ex/ Determine the angle between the vectors:

<1,1>, <2,-2>

,

so .

Determine the angle between the vectors:

 

Determine if the following vectors are orthogonal: