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Chapter 4 : Vector Algebra

4.1 Vectors, what are they anyway ?
4.2 Addition and subtraction
4.3 Magnitude
4.4 Multiplication by a scalar
4.5 Unit vectors
4.6 The dot-product
4.7 The cross-product
4.8 Triple Product
4.9 Summary
4.10 Self-Test, computer programs and problems

4.1 Vectors, what are they anyway ?
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Well, that question is not so easy to answer without getting totally carried away. If you search Encyclopedia Britannica Online for "vector AND mathematics" or for "vector AND physics" you can find the phrase a branch of mathematics that deals with quantities that have both magnitude and direction . From that point of view forces fall under this category. There is a little bit more to it -- that is not everything having direction and magnitude can be treated as a vector as far as applying to it the mathematics of vector algebra and vector analysis is concerned. One test your quantity has to pass is for example whether your quantities can be added to each other like we did when we wanted to find the resultant of forces using Equations 3.2a and 3.2b or Equations 3.4c and 3.4d and whether doing so makes sense (in the case of forces from the view point of physics).

We will see in the next chapter that not only forces but also location vectors -- which are vectors pointing from one point in space to another -- are full-fledged vectors. And so are velocity and acceleration ( EMch 12 stuff).

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4.2 Addition and subtraction
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In section 3.2 and 3.4 we dealt with the subject of finding the resultant of two or more forces :

In the language of mathematics we address the forces , .... etc. as vectors and use the notation :

If we have numerical values available for the components we would write :

The arrow on top of the symbol merely indicates that this quantity is a vector. Many textbooks use bold/italics symbols instead, like F1 because the arrows are costly to print.

On the WWW I am in a similar situation and will use the arrow mode and bold/italics interchangeably.

Using this notation, Equation 3.4c (just above) is then written as

The mathematician would read this as :

Add the vector and to obtain the vector and Equation 3.4c states the rule as to how to do that :

Adding two vectors to each other results in a new vector the x-component of which is determined by the adding the x-components of the given vectors. (Same for the y- and z- component)

The physicist and we would say :

The forces and together have the same action on a given body as the force . To obtain one force from the others we perform the same mathematical operations, that is use Equation 3.4c.

How about subtraction of vectors ?
Formally we would write :

but how would we determine the components of if the components of and are given ? Well, a requirement one could put forward here would be the following :

If you subtract first from a vector a vector to obtain an intermediate vector and than ADD to that, you should come back to . The only way how this can be accomplished by using the following rule :

Subtracting two vectors from each other results in a new vector the x-component of which is determined by subtracting the x-components of the given vectors. (Same for the y- and z- component)

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4.3 Magnitude
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There is not much to say here. We encountered the magnitude of force before and calculated it using Equations 3.2c (for 2-dimensional cases) and 3.4d (for 3-dimensional cases). Just as a reminder :

If the components of a vector are given, its magnitude (which is always non-negative) is calculated according to :

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4.4 Multiplication by a scalar
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A scalar is a single number which can be negative, zero, or positive.

If we wish to multiply a given vector, say by a scalar, say "a", we obtain a new vector, say . We formally write :

and the rule which goes along with that is the following :

Multiplying a vector by a scalar results in a new vector the x-component of which is determined by multiplying the the x-component of the given vector by the scalar. (Same for the y- and z- component)

At least for the case of the scalar being a positive integer this rule makes a lot of sense because it ties in nicely with the addition of vectors : whether you add a given vector 4 times to itself or whether you multiply the vector by 4 should give identical results :

Speaking in terms of forces : 4 times is a force which is 4 times as strong as and pointing in the same direction as .

Similarily, in the following equation the left side and right side give identical results :

Speaking in terms of forces : -1 times is a force which has the same magnitude as but is pointing exactly in opposite direction.

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4.5 Unit vectors
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If F denotes the magnitude of a vector calculated according to Equation 4.3a then we can of course multiply the vector by the scalar 1/F where F=magnitude of . The result is a vector which has a special property and therefore is given in many textbooks a special symbol, the greek lower case lambda :

Because the magnitude F (and with that 1/F) is never negative the vector is pointing in the same direction as the vector itself.
The unique property of is that its magnitude is always 1 (one) regardless of what values (and units) the components of are (except if they are all zero). Because of this property we call a unit vector.

Among the infinitely many unit vectors which one can calculate there are a few worth mentioning. If in Equation 4.5a the vector has a positive x-component but zero y- and z-component, that is if it is pointing along the positive x-axis, then is pointing along the positive x-axis as well, still having the length one. In this special case we often (and this is almost universal) give that unit vector the symbol . The analogs for the y- and z-directions are given the symbols and , respectively.

The unit vectors , , and can be used to present any arbitrary vector with components Fx, Fy, and Fz in an alternative form :

This way of writing out a vector will come in handy later on in the course, for now it is one example of how addition of vectors and multiplication with a scalar appear in a single equation.

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4.6 The dot-product
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The dot-product of two vectors, let's say and is defined as :

in which "F" and "r" denote the magnitude of and , respectively. is the angle enclosed by the two vectors as shown in Figure 4.6a.


Figure 4.6a
Dot-product of two vectors
If you know the two magnitudes and the angle you can use Equation 4.6a to determine the value (which may come out positive, negative, or zero) of the dot-product.

If you know the components of the two vectors (let's say Fx, Fy,Fz, and rx,ry,rz) there is an alternative way to determine the value of the dot-product :

Equations 4.6a and 4.6b can be used nicely to calculate unknown quantities. For example if the components of the two vectors are given and one wishes to know the angle between them, see Prob. 4.9e.
In the 2-dimensional case it is not too difficult to derive Equation 4.6b from 4.6a using only elementary geometry, see Prob. 4.9j.

The vectors and can represent any, even different physical vector quantities.

From the point of view of physics often represents a force acting on a body while represents the distance the same body moves. In this case the value of the dot-product equals the amount of work the force performs on the body.

Here are two equations which come in handy at times :

Another well-known case is when the two vectors and are one and the same. We find that for any vector the dot-product with itself is equal to its magnitude squared :

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4.7 The cross-product
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It would be neat to find out who first formulated the cross-product of vectors, my bet is that it was a physicist.
We will use the cross-product extensively in Chapter 5.


Figure 4.7a
Cross-product of two vectors
Lets assume you have given the components of a force acting on a body of arbitrary shape at some point B as shown in the figure to the left.
The body is supported at point A by what we call a ball/socket joint which allows the body to swivel freely about point A.
Lets call the vector from point A to point B the vector and lets assume we know the value of its components.

Question : About which axis will the given body start to rotate under influence of the force and how effective is the given force in trying to actually rotate the body ?
In Chapter 5 we will do some investigating ourselves, for now I state :

  1. The body will rotate around the axis A-C which stands perpendicular to and .
  2. The strength with which the force tries to rotate the body is measured by a variable M ( which we later will call moment or torque ) the value of which is calculated by the equation :

where is the angle enclosed by the two vectors and .

Well, if I gave you the components of these two vectors you could calculate the magnitudes F and r and use the dot-product to determine the angle between the two vectors and with that the value of M (= moment or torque in physics terms) is determined. The orientation of the axis A-C is harder to determine with the tools we have sofar at our disposal.

This is where the cross-product comes in.

The cross-product of two vectors is a new vector which is perpendicular to the first two vectors and points in such a direction that it gives you also the direction of rotation. The length of this new vector is given by Equation 4.7a.

Formally I write :

but the problem is of course to determine the components of the vector for given and .

If you know what the determinant of a matrix is then the following says it all :

or multiplying it all out :

From ultiplying numbers we know that 2*3 equals 3*2 but, note the following :
On the other hand, the distribute law for scalars and the cross product are identical :

Examples of evaluating the cross-product of given vectors : Prob. 4.9f , and Prob. 4.9g , and Prob. 4.9h .

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4.8 Triple Product
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Assume for the moment that you have three vectors given, like , , and which from the point of view of mathematics can represent anything (not just forces). The triple product between these three vectors is defined as :

The parenthesis indicate that you first evaluate the cross-product, which results in a new vector, and then evaluate the dot-product which results in a scalar "d".

An alternate way to determine the value of "d" is to evaluate the following determinant :

A property of determinants is that exchanging any two rows will merely change the sign (+/-) of the answer. As a result you can change the order of the vectors in all together 6 different ways of which I am just showing three :

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4.9 Summary
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In preparations for things to come ( i.e. use of the cross-product in the next chapter ) I summarized some of the rules of vector algebra. Specifically, have been defined and hopefully put into context. More will be said as we actually apply the developed equations.

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4.10 Self-Test and problems
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Self-Test

The self-test is a multiple-choice test. It allows you to ascertain your knowledge of the definition of terms and your understanding of important results.

Click here to do the test.

Computer Programs

None available for this chapter. Sorry.

Problems

Please, try them all.

Prob. 4.10a : Addition/subtraction of vectors
Prob. 4.10b : Multiplication by a scalar
Prob. 4.10c : Magnitude of Unit vector
Prob. 4.10d : Value of dot product
Prob. 4.10e : Dot product -- Angle between vectors
Prob. 4.10f : Evaluate cross product
Prob. 4.10g : Proof on cross product
Prob. 4.10h : Cross product -- Angle between vectors
Prob. 4.10i : Broadcast pole, 3-D (Prob 3.6h revisited)
Prob. 4.10j : Proof on dot-product

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Zig Herzog, hgn@psu.edu

Last revised: 09/29/10