Vectors and Vector Algebra
Study Guide for the online tutorial Vector Algebra developed by the University of Oklahoma.
http://www.nhn.ou.edu/~walkup/demonstrations/WebTutorials/VectorIntroduction.htm
Answer the questions or fill in the blanks in this study guide as you go through the Vectors
and Vector Algebra tutorial. Take screen shots as requested (Fn F11) and paste into this
document.
There will be a quiz on Monday, Dec 7, on the concepts covered in this tutorial.
Why do we bother with vector algebra?
Learning vector algebra represents an important step in students' ability to solve problems. The importance of
vector algebra can be understood in the context of previous steps in knowledge:

At some point (usually in middle school or high school) students are taught basic algebra because the
mathematics they have known up to that point, arithmetic, cannot solve _________. For example, a
student may be asked to find the speed required to travel 33 miles in 60 minutes. For this problem,
arithmetic alone is not terribly useful.

During high school students begin to realize that even algebra cannot solve problems that incorporate
__________, so they learn trigonometry and geometry. For example, if a student was trying find the
amount of concrete needed to fill a cone-shaped hole, simple algebra alone will be of little help.

However, geometry and trigonometry are very difficult to apply in many situations. __________ was
invented in order to solve __________ problems without the use of cumbersome geometry.
Although it is possible to use ordinary trigonometry and geometry to solve most of the physics problems you
are likely to encounter, vector algebra has some significant advantages:
1. Vector algebra is __________ than geometry and requires knowledge of fewer rules.
2. The mechanics of vector algebra are __________, requiring less intuition and cleverness in finding a
solution. (Remember those nasty geometry proofs from high school?)
3. Vector algebra operations are much __________ with familiar __________. (For example, the
statement C = A + B is a typical vector algebra expression.)
4. Many of the rules learned in __________ also apply in vector algebra. (For example, you can add the
same vector to both sides of an equation, you can divide both sides of an equation by a number, and so
on.)
What is a vector?
Suppose you were given the job as weatherman for your local television station. You would like to convey to
your audience the wind speeds and directions in their area, and how they compare to other areas. First you
could try writing down the speed and direction of the wind at various locations on a map.
However, notice that the viewer is going to have a miserable time. First, he will need very good eyesight,
and patience. Most importantly, such a system is insufficient for detecting general trends in __________
and __________. To do so, the viewer has to write down all of the data for each location and study it -- not
a good system!
A Slight Improvement
Another idea is to indicate the direction of the wind with an arrow, while writing the wind speed next to the
arrow. Now the viewer can compare the directions of the wind easily with other locations.
Some of the arrows may correspond to mere breezes, others to full-blown gales. Therefore, such a system is
__________, because numbers alone do not provide the necessary visual cues to establish general trends in
wind speed.
Using the Idea of the Vector
Is there a way that we can incorporate both aspects of wind, speed and direction, with a simple system?
Yes! Make the length of the arrow correspond to the speed of the wind. With this system, long arrows
correspond to high winds, and so on. Now, the viewer can tell which direction the wind is blowing in his area
with a quick glance. Furthermore, the viewer can tell how the wind speed in his area compares other areas.
Once the arrow indicates both a __________ of some sort (in this case, the speed of the wind) and a
__________, it is called a __________. The vector in this example is a __________. The length of the arrow,
which represents the magnitude of the velocity, is called the speed. Notice that speed and velocity are not
synonyms in physics -- the term velocity refers to a __________ quantity and has both a magnitude (the speed)
and direction.
Notice that if the viewer wants to know exactly how strong the wind is in a particluar location he will still have
to refer to numerical data -- the length of the vector arrow is not sufficiently precise to provide this information.
So at this point, we would guess that vectors have very limited __________ use.
But we would be __________.
Vector Algebra
What is vector algebra?
As we learned on the previous page, vectors __________ have limited use other than providing a simple, yet
effective, means of displaying quantities possessing both a magnitude and direction. The real power in vectors
resides in the ability to perform __________ on them.
An algebra is a set of __________. And in order to use vector algebra, you have to know the rules. Fortunately
for life science majors (and high school students), there is only one rule you have to remember -- the rule for
__________. (We will see that the operation of subtraction is essentially the same as ___________. And if you
can add two vectors together, then adding three or more vectors is straightforward.)
Those studying a calculus-based physics course also have to consider how to multiply vectors, but we will not
concern ourselves with this added burden. So vector algebra is actually simpler than regular algebra because we
only have to concern ourselves with one operation -- addition.
Some Important Points about Notation and Definitions
We need notation for labeling vectors: A vector is denoted by a bold-face letter, and its length is denoted by the
same letter without the bold face.
B
B
A vector, with magnitude and
direction
The magnitude of the vector B.
We now need to introduce a definition: The sum of two or more vectors is another vector called the
__________.
Vector addition for two vectors A and B is simply denoted A + B. Therefore, the equation C = A + B simply
means "C is the resultant vector obtained by adding vector A to vector B." Vector subtraction of vector B from
vector A is simply denoted A - B.
There are three ways to add vectors. Each method will be illustrated using Java applets in the following pages.
Naturally, all three methods must produce the same result.

The head-to-tail method.

The parallelogram method.

The component method.
The head-to-tail and parallelogram methods are actually identical, as a Java applet will later demonstrate. They
only provide a rough description of the resultant vector, but they are very easy to apply. The component
method is used in those situations where exact, numerical information about the resultant vector is required.
The Head-to-Tail Method
FAQ: What are the properties of a vector?
What CANNOT be done to the vector when moving it?
You might want to make one vector in the x-direction and the second one at an angle between 20 and 80
degrees. Move the second vector’s tail to the head of the first vector and see the result of adding the vectors.
Take a screen shot of your first vector addition and paste it here. Repeat for subtracting vectors. Remove the
second vector and draw a similar one only with the head of the vector at the origin, then move the head to the
head of the first vector. Take a screen shot and paste it here.
Special Examples
Take a screen shot of your collinear vector addition and one of the collinear subtraction.
Multiple Vector Summation
Try several examples with 3 vectors and take screen shots of your work showing the resultant.
Notice one distinct disadvantage of this method -- finding the exact length and direction of the resultant vector
is usually going to require some sophisticated geometry, especially when summing multiple vectors. Therefore,
this method is used mainly to provide a __________ description of the sum of one or more vectors.
Parallelogram Method—Omit
Vector Subtraction—Omit
The Component Method, Part 1
The Basics - Good Vectors from Bad
The component method of summing vectors is universally feared by introductory physics students, but is
actually simple as long as you don't get too worried about trigonometric details. The foundation of the
component method actually relies on a basic principle:
Vectors are easy to sum if they fall into two categories:


The vectors point along the __________. In this case, the sum of the two vectors is just the
summation of the lengths if they point in the same direction, and the subtraction of the two lengths if
they point in opposite directions.
The vectors are __________ to each other. Here, simple mathematical relationships can be used to
solve for the resultant vector.
Omit web assignment.
What are component vectors?
Consider the equation vector C = A + B, which simply states that "vector C is the sum of vectors A and B."
This statement is an equality, which is a very strong statement -- it means that C can be replaced with the term
A + B whenever we see fit.
We will take advantage of this property to greatly simplify vector addition by replacing vectors that are hard to
sum (we will call them __________) with vectors that are easy to sum (__________).
Now there are any number of vectors that sum to vector C, but we are going to choose A and B very carefully.
Consider the following example, where we are trying to add the two vectors C and G. You should appreciate
how difficult it is to add these two vectors because they do not point along the same direction, nor are they
perpendicular. Therefore, we consider them bad vectors.
But why not replace both bad vectors with good vectors that are easy to sum?
In the figure below, we have defined two directions with dashed lines we call the x-direction (horizontal) and ydirection (vertical). If we can somehow find replacement vectors for C and G that line up along these
directions, then we will be guaranteed that all vectors will either be pointing along the same direction or
perpendicular to each other.
But how do we find these good vectors?
Take a screen shot of the applet showing a vector in Quadrant III and paste it here.
A Little Bit of Trig (and I Mean Little)
Now that we know we have to replace C with A and B, how do we find the lengths of A and B? (We already
know their directions, right?) To find these lengths is simple, and requires nothing more than the ability to
punch the "COS" and "SIN" buttons on your calculator.
But what are COS and SIN?
Notice in the applet above that vectors A and B are always __________ than C. The COS and SIN buttons
simply take the length and direction of C and use this information to find the lengths of A and B. In the figure
below we examine our earlier example, defining a direction of C of 58 degrees with respect to the x-direction
(horizontal).
But which do I choose? COS or SIN?
The rule here is simple: Do you see the angle symbol that we have drawn in the figure? Well, if that angle sign
touches the component vector for which you are trying to find its length, use COS.
If the angle sign touches, it's COS!
If you use COS on one component vector, use SIN for the other. In summary, COS and SIN are "shrink
factors" -- the amount of shrinkage depends on the angle.
So in our example above, the length of vector B is given by C X COS(58). The reason? The angle symbol
touches vector B, so we use the COS button. Since we used COS for B, we use SIN for A. In summary,
B = C X COS(58),
A = C X SIN(58).
Why are the vector symbols not boldfaced in the equations above? (Link doesn’t work)
What do the SIN and COS buttons really mean? Answer here:
When you get to the Quiz Tool, first ask for a new question and take a screen shot of your correct answer and
paste it here. Repeat for Quiz Tool 2.
Dropbox the file as LastNamesVectorTutorial.doc.