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Chapter 1. Vectors
We are all familiar with the distinction between things which have a direction and those which don't. The velocity of the wind (see figure 1.1) is a classical example of a vector
quantity. There are many more of interest in physics, and in this and subsequent chapters we will try to exhibit the fundamental properties of vectors. Vectors are intimately related to the very nature of space. Euclidian geometry (plane and spherical geometry) was an early way of describing space. All the basic concepts of Euclidian geometry can be expressed in terms of angles and distances. A more recent development in describing space was the introduction by Descartes of coordinates along three orthogonal axes. The modern use of Cartesian vectors provides the mathematical basis for much of physics.
A. The Displacement Vector
The preceding discussion did not lead to a definition of a vector. But you can convince yourself that all of the things we think of as vectors can be related to a single fundamental quantity, the vector r representing the displacement from one point in space to another. Assuming we know how to measure distances and angles, we can define a displacement vector (in two dimensions) in terms of a distance (its magnitude), and an angle:
12
displacement from point 1 to point 2 distance, angle measured counterclockwise from due East
r
(See figure 1.2.) Note that to a given pair of points corresponds a unique displacement, but a given displacement can link many different pairs of points. Thus the fundamental definition of a displacement gives just its magnitude and angle. We will use the definition above to discuss certain properties of vectors from a strictly geometrical point of view. Later we will adopt the coordinate representation of vectors for a more general and somewhat more abstract discussion of vectors.
Figure 1-1. Where is the vector?
B. Vector Addition
A quantity related to the displacement vector is the position vector for a point. Positions are not absolute – they must be measured relative to a reference point. If we call this point O (the "origin"), then the position vector for point P can be defined as follows:
rP displacement from point O to point P (1-2)
It seems reasonable that the displacement from point 1 to point 2 should be expressed in terms of the position vectors for 1 and 2. We are be tempted to write r 12 (^) r 2 r 1
^ (1-3)
A "difference law" like this is certainly valid for temperatures, or even for distances along a road, if 1 and 2 are two points on the road. But what does subtraction mean for vectors? Do you subtract the lengths and angles, or what? When are two vectors equal? In order to answer these questions we need to systematically develop the algebraic properties of vectors.
We will let A
, B
, C
, etc. represent vectors. For the moment, the only vector quantities we have defined are displacements in space. Other vector quantities which we will define later will obey the same rules.
Definition of Vector Addition. The sum of two vector displacements can be defined so as to agree with our intuitive notions of displacements in space. We will define the sum of two displacements as the single displacement which has the same effect as carrying out the two individual displacements, one after the other. To use this definition, we need to be able to calculate the magnitude and angle of the sum vector. This is straightforward using the laws of plane geometry. (The laws of geometry become more complicated in three dimensions, where the coordinate representation is more convenient.)
Let A
and B
be two displacement vectors, each defined by giving its length and angle:
B
A B B
A A
point 1
point 2
r
east
angle
distance
Figure 1-2. A vector, specified by giving a distance and an angle.
cosines: C^2 = A^2 + B^2 -2AB cos 2 giving C = [(10 m)^2 + (14 m)^2 - 2(10m)(14m)cos 152]1/ = 23.3072 m. The angle 1 can be calculated from the law of sines: sin 1 / B = sin 2 / C giving 1 = sin-1^. = 16.380 . The angle C is then equal to 48 - 1 = 31.620 . The result is thus
C (^) 23.3072 m, 31.620.
One conclusion to be drawn from the previous example is that calculations using the geometrical representation of vectors can be complicated and tedious. We will soon see that the component representation for vectors simplifies things a great deal.
C. Product of Two Vectors
Multiplying two scalars together is a familiar and useful operation. Can we do the same thing with vectors? Vectors are more complicated than scalars, but there are two useful ways of defining a vector product.
The Scalar Product. The scalar product, or dot product, combines two vectors to give a scalar:
A B AB c os ( B - A)
48
20
1
2
3
A
B
C
10 m
14 m
C
C A B
48 20
1
2
3
48 - 20 = 28
180 - 28 = 152
Figure 1-4. Example of vector addition. Each vector's direction is measured counterclockwise from due East. Vector A
is a displacement of 10 m at an angle of 48and vector B
is a displacement of 14 m at an angle of 20.
This simple definition is illustrated in figure 1-5. One special property of the dot product
is its relation to the length A of a vector A
A A A^2 (1-6)
This in fact can be taken as the definition of the length, or magnitude, A.
An interesting and useful type of vector is a unit vector, defined as a vector of length 1. We usually write the symbol for a unit vector with a hat over it instead of a vector symbol: u ˆ. Unit vectors always have the property u ˆ^ u ˆ 1 (1-7) Another use of the dot product is to define orthogonality of two vectors. If the angle between the two vectors is 90, they are usually said to be perpendicular, or orthogonal. Since the cosine of 90 is equal to zero, we can equally well define orthogonality of two vectors using the dot product:
A B A B 0 (1-8)
Example. One use of the scalar product in physics is in calculating the work done by a force F
acting on a body while it is displaced by the vector displacement d
. The work done depends on the distance moved, but in a special way which projects out the distance moved in the direction of the force: Work F d (1-9) Similarly, the power produced by the motion of an applied force F
whose point of application is moving with velocity v is given by Power F v (1-10) In both of these cases, two vectors are combined to produce a scalar.
Example. To find the component of a vector A in a direction given by the unit vector n ˆ, take the dot product of the two vectors. Component of A along n A n ˆ (1-11)
The Vector Product. The vector product, or cross product, is considerably more complicated than the scalar product. It involves the concept of left and right, which has an interesting history in physics. Suppose you are trying to explain to someone on a distant planet which side of the road we drive on in the USA, so that they could build a car, bring it to San Francisco and drive around. Until the 1930's, it was thought that there was no way to do this without referring to specific objects which we arbitrarily designate as left-handed or right-handed. Then it was shown, by Madame Wu, that both the electron and the neutrino are intrinsically left-handed! This permits us to tell the alien how to determine which is her right hand. "Put a sample of 60 Co nuclei in front of you, on a mount where it can rotate freely about a vertical axis. Orient the nuclei in a
A
B
B - A
Figure 1- 5. Vectors A
and B
. Their dot product is equal to A B cos(B - A). Their cross product A B
has magnitude A B sin(B - A), directed out of the paper.
Example. The cross product is used to find the direction of the third axis in a three-dimensional space. Let u ˆand v ˆbe two orthogonal unit vectors, representing the first (or x ) axis and the second (or y ) axis, respectively. A unit vector w ˆ in the correct direction for the third (or z ) axis of a right-handed coordinate system is found using the cross product: w ˆ u ˆ v ˆ (1-15)
D. Vectors in Terms of Components
Until now we have discussed vectors from a purely geometrical point of view. There is another representation, in terms of components, which makes both theoretical analysis and practical calculations easier. It is a fact about the space that we live in that it is possible to find three, and no more than three, vectors that are mutually orthogonal. (This is the basis for calling our space three dimensional.) Descartes first introduced the idea of measuring position in space by giving a distance along each of three such vectors. A Cartesian coordinate system is determined by a particular choice of these three vectors. In addition to requiring the vectors to be mutually orthogonal, it is convenient to take each one to have unit length.
A set of three unit vectors defining a Cartesian coordinate system can be chosen as
follows. Start with a unit vector i ˆin any direction you like. Then choose any second
unit vector ˆ j which is perpendicular to i ˆ. As the third unit vector, take k ˆ^ i ˆ j ˆ.
These three unit vectors (ˆ i ,ˆ j , k ˆ)are said to be orthonormal. This means that they are
mutually orthogonal, and normalized so as to be unit vectors. We will often refer to their directions as the x, y, and z directions. We will also sometimes refer to the three vectors as ( e ˆ 1 , e ˆ 2 , e ˆ 3 ), especially when we start writing sums over the three directions.
Suppose that we have a vector A
and three orthogonal unit vectors ( i ˆ,ˆ j , k ˆ), all defined
as in the previous sections by their length and direction. The three unit vectors can be
used to define vector components of A
, as follows:
A A k
A A j
A A i
z
y
x
This suggests that we can start a discussion of vectors from a component view, by simply defining vectors as triplets of scalar numbers:
x y z
A
A A
A
^
Component Representation of Vectors (1-17)
It remains to prove that this definition is completely equivalent to the geometrical definition, and to define vector addition and multiplication of a vector by a scalar in terms of components.
Let us show that these two ways of specifying a vector are equivalent - that is, to each geometrical vector (magnitude and direction) there corresponds a single set of components, and (conversely) to each set of components there corresponds a single geometrical vector. The first assertion follows from the relation (1-16), showing how to determine the triplet of components for any given geometrical vector. The dot product of any two vectors exists, and is unique.
The converse is demonstrated in Figure 1-7. It is seen that the vector A
can be written as the sum of three vectors proportional to its three components:
A iAx jAy kAz
From the diagram it is clear that, given three components, there is just one such sum. So, we have established the equivalence
x y z
A
A magnitude direction A A
^
E. Algebraic Properties of Vectors.
As a warm-up, consider the familiar algebraic properties of scalars. This provides a road map for defining the analogous properties for vectors. Equality.
x
y
z
iAx ˆ jAy ˆ
kAz ˆ
Figure 1 - 7. Illustration of the addition of the component vectors i Ax, j Ay, and k Az to get the vector A
. This proves that a given set of values for (Ax, Ay, Az) leads to a unique vector A
in the geometrical picture.
Multiplication of a vector by a scalar. We will take the following, rather obvious,
definition of multiplication of a vector A
by a scalar c : x y z
cA cA cA cA
^
DEFINITION (1-23)
It is pretty clear that this is consistent with the procedure in the geometrical representation: just multiply the length by the scalar c, leaving the angle unchanged.
The Zero Vector
We define the zero vector as follows:
0 0 0 0
^
DEFINITION (1-24)
Taken with the definition of vector addition, it is clear that the essential relation
A 0 A
y
z
x
A
B
C
Figure 1.7a. The components of the sum vector C are seen to be the algebraic sum of the components of the two vectors being summed.
is satisfied. And a vector with all zero-length components certainly fills the bill as the geometrical version of the zero vector.
The Negative of a Vector. The negative of a vector in terms of components is also easy to guess:
x y z
A
A A
A
^
DEFINITION (1-26)
The essential relation A A 0 will clearly be satisfied, in terms of components. It is
also easy to prove that this corresponds to the geometrical vector with the direction reversed; we will also omit this proof.
Subtraction of Vectors. Subtraction is then defined by
A B A ( B ) subtraction of vectors (1-27)
That is, to subtract a vector from another one, just add the vector's negative. The "vector-subtraction parallelogram" for
two vectors A and B is shown in figure 1-8. The challenge is to choose the
directions of A and B such that the diagonal correctly represents head-to-tail addition of the vectors on the sides.
E. Algebraic properties of vector addition.
Vectors follow algebraic rules similar to those for scalars:
commutative property of vector addition ( ) ( ) ) associative property of vector addition distributive property of scalar multiplication
another
A B B A
A B C A B C
a A B aA aB
a b A aA aB
^
distributive property
c dA cd A associative property of scalar multiplication
In the case of geometrically defined vectors, these properties are not so easy to prove, especially the second one. But for component vectors they all follow immediately (see the problems). And so they must be correct also for geometrical vectors.
As an illustration, we will prove the distributive law of scalar multiplication, above, for component vectors. We use only properties of component vectors.
B A
A
A
B
B
A B
Figure 1-8. The vector-subtraction parallelogram. Can you put arrows on the sides of the parallelogram so that both triangles read as correct vector-addition equations?
Notice that in the preceding box, vectors are not specifically defined. Nor is the method of adding them specified. We will see later that there are many different classes of objects which can be thought of as vectors, not just displacements or other three- dimensional objects.
Example: Check that the set of all component vectors, defined as triplets of real
numbers, does in fact satisfy all the requirements to constitute a vector space.
Referring to Table 1-1, it is easy to see that the first four properties of a vector space are satisfied:
- Closure under addition. If
x y z
A
A A
A
and
x y z
B
B B
B
are both vectors, then so
is
x x y y z z
A B
C A B A B
A B
^
. This follows from the fact that the sum of two scalars
gives another scalar.
A vector space is a set of objects, called vectors, with the operations of addition of two vectors and multiplication of a vector by a scalar defined, satisfying the following properties.
- Closure under addition. If A
and B
are both vectors, then so is C A B.
- Closure under scalar multiplication. If A
is a vector and d is a scalar, then B dA is a vector.
- Existence of a zero. There exists a zero vector 0
, such that, for any vector A
A A
- Existence of a negative. For any vector A
there exists a negative A
, such that ( ) 0
A A ^.
- Algebraic Properties. Vector addition and scalar multiplication satisfy the following rules:
(1-29) commutative ( ) ( ) (1-30) associative ( ) (1-31) distributive (1-32) distr
A B B A
A B C A B C
a A B aA aB a b A aA bA
ibutive
c dA ( cd A ) (1-33) associative
Table 1-1. Properties of a vector space.
- Closure under multiplication. If A
is a vector and d is a scalar, then x y z
dA B dA dA dA
^
is a vector. This follows from the fact that the product of two
scalars gives another scalar.
- Zero. There exists a zero vector
, such that, for any vector A
x x y y z z
A
A A A
A
. This follows from the addition-of-zero property for
scalars.
- Negative. For any vector A
there exists a negative
x y z
A
A A
A
^
, such that
A A . Adding components gives zero for the components of the sum.
- The algebraic properties (1-29) through (1-33) were discussed above; they are satisfied for component vectors.
So, all the requirements for a vector space are satisfied by component vectors. This had better be true! The whole point of vector spaces is to generalize from component vectors in three-dimensional space to a broader category of mathematical objects that are very useful in physics.
Example: The properties above have clearly been chosen so that the usual
definition of vectors, including how to add them, satisfies these conditions. But the concept of a vector space is intended to be more general. What if we define vectors in two dimensions geometrically (having a magnitude and an angle) and we keep multiplication by a scalar the same, but we redefine vector addition in the following way.
C A B ( A B , (^) A B ) (1-34)
This might look sort of reasonable, if you didn't know better. Which of the properties (1)-(5) in Table 1-1 are satisfied?
1. Closure under addition: OK. A + B is an acceptable magnitude, and A +
B is an acceptable angle. (Angles greater than 360 are wrapped around.)
- Closure under scalar multiplication: OK
- Zero: the vector (^0) (0,0) works fine; adding it on doesn't change A.
- Negative: Not OK. There is no way to add two positive magnitudes (magnitudes are non-negative) to get zero.
- Algebraic properties: You can easily show that these are all satisfied.
A
A
A k A
A
A
A j A
A
A
A i A
z
y
x
cos^1 ˆ
cos^1 ˆ
cos^1 ˆ
Specifying these three values is one way of giving the direction of a vector. However, only two angles are required to specify a direction in space, so these three angles must not be independent. It can be shown (see problems) that
cos 2 cos^2 cos^2 1
Definition of a Metric Space. The properties given in table 1-1 constitute the standard definition of a vector space. However, inclusion of a scalar product turns a vector space into the much more useful metric space, as defined in table 1-2. The difference between a vector space and a metric space is the concept of distance introduced by the inner product.
H. The vector product.
The geometrical definition of the cross product of A
and B
results in a third vector, say, C
. The relation between A
, B
and is C
quite complicated, involving the idea of right- handedness vs. left-handedness. We have already built a handedness into our coordinate
system in the way we choose the third unit vector, k ˆ^ i ˆˆ j. As a preliminary to
A metric space is defined as a vector space with, in addition to its usual properties, an inner product defined which has the following properties:
- If A
and B
are vectors, then the inner product (^) A B is a scalar.
- A A 0 0
A . (1-44)
- The following algebraic properties of the inner product must be obeyed: A B B A
A B C A B A C
A aB a A B
Table 1-2. Properties of a metric space. Note that the scalar product of two vectors as just defined has all of these properties.
A
x
y
z
Figure 1 - 9. The direction cosines for the vector A are the cosines of the three angles shown.
evaluating the cross product A^ B
, we work out the various cross products among the
unit vectors i ˆ (^) , j ˆ, k ˆ. From equation (1-17) we see that the cross product of two
perpendicular unit vectors has magnitude 1. We use the right-hand rule and refer to figure 1-11 to get the direction of the cross products. This gives
i k j
k j i
j i k
k i j
j k i
i j k
ˆ^ ˆ
Now it is straight forward to evaluate A B
in terms of components: ˆ ˆ ˆ ˆ ˆ ˆ ;
ABk AB j ABk ABi AB j AB i
A B iA jA kA iB jB kB
x y x z y x y z z x z y
x y z x y z
This is not as hard to memorize as you might think - stare at it for a while and notice permutation patterns ( yz vs. zy, etc.) Later on we will have some other, even more elegant ways of writing the cross product.
The cross product is used to represent a number of interesting physical quantities - for
example, the torque, r F
, and the magnetic force, F qv B
, to name just a
couple.
The cross product satisfies the following algebraic properties:
(1-48) ( ) (1-49) ( ) ( ) (1-50)
A B B A
A B C A B A C
A aB a A B
Note that the order matters; the cross product is not commutative.
I. Dimensionality of a vector space and linear independence.
In constructing our coordinate system we used a very specific procedure for choosing the directions of the axes, which only works for 3 dimensions. There is a broader, general question to be asked about any vector space: What is the minimum number of vectors required to "represent" all the others? This minimum number n is the dimensionality of a vector space.
y z z y
z x x z
x y y x
A B i A B A B
j A B A B
k A B A B
definition of the cross product
^
cE cE cE . (1-53)
This means that the sums of x-components, y-components and z-components must separately add up to zero, giving three equations:
3 0 z-component
3 2 0 y-component
2 4 3 0 x component
1 2 3
1 2 3
1 2 3
c c c
c c c
c c c .
Now we solve for c 1 , c 2 , and c 3. Subtracting the third equation from the second gives c 3 =0. The first and second equations then become
1 2
1 2
c c
c c .
The first equation gives c 1 = -2c 2 , and the second equation gives c 1 = -3c 2. The only consistent solution is c 1 = c 2 = c 3 = 0. These three vectors are linearly independent!
This second example is a little messier and less satisfying than the previous example, and it is clear that in 4, 5 or more dimensions the process would be difficult. In chapter 2 we will discuss more elegant and powerful methods for solving simultaneous linear equations.
Solving simultaneous linear equations with Mathematica. It is hard to resist asking Mathematica to do this problem. Here is how you do it:
Solve[{2c1+4c2+3c3==0,c1+3c2+2c3==0,c1+3c2+c3==0},{c2,c3}] {}
This is Mathematica’s way of telling us that there is no solution.
What if we try to make E 1
, E 2
, and E 3
linearly dependent? If we change E 2
to
then the sum of E 1
and E 2
is twice E 3
, so the linear dependence relation
1 2 2 3 0
E E E
should be satisfied; this corresponds to c 2 = c 1 , c 3 = -2c 1. Let’s ask Mathematica:
Solve[{2c1+4c2+3c3==0,c1+3c2+2c3==0,c1+c2+c3==0},{c2,c3}] {{c2->c1,c3->-2 c1}}
Sure enough!!
Is this cheating? I don't think so!
J. Components in a Rotated Coordinate System. In physics there are lots of reasons for changing from one coordinate system to another. Usually we try to work only with coordinate systems defined by orthonormal unit vectors. Even so, the coordinates of a vector fixed in space are different from one such coordinate system to another. As an
example, consider the vector A
shown in figure 1-11.
It has components Ax and Ay, relative to the x and y axes. But, relative to the x' and y' axes, it has different components Ax' and Ay'. You can perhaps see from the complicated construction in figure 2-3 that
Ax'=Ax cos + Ay sin (1-54)
And a similar construction leads to
Ay'=-Ax sin + Ay cos (1-55)
There is another, easier (algebraic rather than geometrical) way to obtain this result:
ˆ ˆ' ˆ ˆ'
A i i A j i
iA jA i
A A i
x y
x y
x
,
ˆ ˆ' ˆ ˆ'
A i j A j j
iA jA j
A A j
x y
x y
y
,
We can evaluate the dot products between unit vectors with the aid of figure 1-12, with the result
x
y
A
Ax
x'
y'
Ay
Ax'
Ax cos
Ay sin
Figure 1-11. Coordinates of a vector A
in two different coordinate systems.
i
i'
j' j
Figure 1 - 12. Unit vectors for unrotated ( i and j ) and rotated ( i' and j' ) coordinate systems.
