This module is part of a series of modules designed for teaching the physics component of GAME2302 Mathematical Applications for Game Development at Austin Community College in Austin, TX. (See GAME 2302-0100: Introduction for the first module in the course along with a description of the course, course resources, homework assignments, etc.)

The purpose of this module is to introduce motion and to explain displacement and vector analysis.

I recommend that you open another copy of this document in a separate browser window and use the following links to easily find and view the Images and Listings while you are reading about them.

As mentioned above, if one body is moved with respect to another body, either one of them may be said to have experienced a displacement .

Displacement is not the same as distance

Assume, for example, that you go to the local high school and run five laps around the track, stopping exactly where you started. The distance that you have run will be the length of the path that you followed while running. However, while you may be completely out of breath by that point in time, the magnitude of your displacement will have been zero.

Scalars versus vectors

Distance is a scalar quantity, while displacement is a vector quantity. Scalar quantities have only magnitude, such as three kilometers.

Vector quantities have two parts: magnitude and direction . For example, you might describe a vector quantity as having a magnitude of three kilometers and a direction of northwest.

Magnitude and direction

A displacement vector must have both magnitude and direction. For example, if you were to run about three-fourths of the way around the track, the distance that you ran would be the length of your path.

However, the magnitude of your displacement would be the straight-line distance from your starting point to your stopping point.

The direction of your displacement would be an angle measured between some reference (such as due east, for example) and an imaginary line connecting your starting point and your stopping point.

If you stop where you start...

Getting back to the original proposition, if your stopping point is exactly the same as your starting point, the magnitude of your displacement vector will be zero and the angle of your displacement vector will be indeterminate.

Vector notation

Assume that there are three spots marked on the top of a table as A, B, and C. If an object is moved from A to B and then from B to C, using vectors, it may be stated that:

vecAC = vecAB + vecBC

where vecAB, vecBC, and vecAC represent vectors.

Typical notation

A typical notation for a vector in a physics textbook is something like AB, possibly in boldface and/or italics, with a small arrow drawn above the two letters. However, there is no such small arrow on a QWERTY keyboard, making it very difficult for me to reproduce in these modules.

My alternative notation

Therefore, in these physics modules, I will use the following notation

vecAB

to indicate a vector extending from the spot labeled A to the spot labeled B.

In order to determine direction in an unambiguous way, you often need to know where the vector starts (the tail) and where it ends (the head). The notation used above indicates that the tail of the vector is at A and the head is at B.

In some cases, particularly in JavaScript code where a vector is defined by magnitude and direction instead of by a starting point and an ending point, I will refer to a vector simply as

vecA, vecB, etc.

A graphic illustrating displacement

Image 1 is a graphic that illustrates displacement.

1

Image 1: A graphic showing displacement.

The triangle shown in Image 1 is a physical representation of the vector equation

vecAC = vecAB + vecBC

As usual, please ignore the lower-case letters scattered throughout the image, which were placed there for a purpose other than for use in this module.

What does this mean?

This vector equation does not mean that the algebraic sum of two sides of the triangle are equal to the third side. However, it does mean that the displacement vecAB together with the displacement vecBC, have an effect that is equivalent to the single displacement vector vecAC.

Scalar addition and subtraction

When adding or subtracting scalars, you only need to take the magnitude values of the scalars into account and perform normal arithmetic using those magnitude values. An example would be to add 6 birds and 5 birds to produce a sum of 11 birds.

Vector addition and subtraction

However, when adding or subtracting vectors, you not only need to take the magnitude of the vectors into account, you also need to take the directions of the vectors into account.

A thought exercise

Image 2 shows the vector diagram for a familiar example.

2

Image 2: Vector diagram for a familiar example.

Pretend that you are standing at point A in Image 2 . Then pretend that you walk 30 meters due east and designate that location as point B.

Up to this point, your displacement can be represented by vecAB with a magnitude of 30 and an angle of zero (assuming that due east represents an angle of zero).

Walk 40 meters due north

Now pretend that you walk 40 meters due north and mentally designate the stopping point as C. Your displacement relative to point B can be represented as vecBC with a magnitude of 40 meters and a direction of north or 90 degrees.

Displacement relative to point A

The interesting question is, what is your displacement relative to your starting point at A?

The displacement vector vecAC is the straight line from point A to point C, pointing in a direction from A to C. This vector is shown in the vector equation

vecAC = vecAB + vecBC

as the sum of the two vectors that describe the two segments of your journey from A to C.

Graphical addition of vectors

This process of connecting two or more vectors tail-to-head in sequence and then measuring the distance and direction from the tail of the first vector to the head of the last vector is known as the graphical addition of vectors . The resulting vector (in this case vecAC) is known as the sum of the vectors that make up the sequence. (The resulting vector is often called the resultant.)

A 3-4-5 triangle

Recalling what we learned in an earlier module where I discussed a 3-4-5 triangle, the magnitude of the vector from A to C (the hypotenuse of a right triangle with sides of 30 and 40) is 50 meters. You may also recall that the angle is approximately 53 degrees relative to the horizontal (east-west) line.

You should be able to measure the length of vecAC as well as the angle between that vector and the horizontal in Image 2 and get reasonable agreement with the above.

No requirement for a right triangle

While this example is based on a right triangle, that is not a requirement for adding vectors. I chose a 3-4-5 right triangle for the example because I knew what the answer would be in advance and also because it should have been familiar to you.

Graphic representation of vectors

Many problems in kinematics and kinetics can be solved graphically using a graphical representation of vectors.

Vectors are typically drawn as a heavy line between two points with an arrow head at one end and nothing in particular at the other end. The end with the arrow head is commonly called the head of the vector and the other end is commonly called the tail.

Somewhat inconvenient

Drawing vectors on graph paper can be inconvenient, particularly if you don't have graph paper available. Fortunately, as you will soon learn, it isn't necessary to use graphics to solve vector problems. You can also solve such problems mathematically, which will often be the better choice.

There are at least two kinds of quantities in physics:

You already know how to do scalar arithmetic. Otherwise, you probably wouldn't be interested in physics.

Another example

Let's go back to our original equation

vecAC = vecAB + vecBC

and assume that the magnitude of vecAB is 30 and the magnitude of vecBC is 40. The sum of those two vectors (vecAC) can have a magnitude ranging anywhere from 10 to 70 depending on the relative angles of vecAB and vecBC.

A triangle with sides of 30, 4 0, and ?

Consider the triangle shown in Image 3 .

3

Image 3: A triangle with sides of 30, 40, and ?

Pretend that instead of walking due north from point B as shown in Image 2 , you change direction and walk northwest (135 degrees relative to the east-west horizontal line with east being zero degrees) keeping the length at 40 meters.

What happened to the displacement?

What is the displacement of the point C relative to the point A? I can't do the arithmetic in my head for this problem, but I can measure the length of vecAC to be about 28 meters and the angle of vecAC to be a little less than 90 degrees. (I will show you how to write a script to solve this problem mathematically later.)

Any number of displacements can be added

There is no limit to the number of displacement vectors that can be added in this manner. For example, pretend that you walk from point A,

Your displacement vecAG will be

vecAG = vecAB + vecBC + vecCD + vecDE + vecEF + vecFG

even if your zigzag path crosses back over itself one or more times.

Vector diagram for the sum of six vectors

Image 4 shows the graphical addition of the six vectors described above.

4

Image 4: Vector diagram for the sum of six vectors.

The displacement in Image 4 is the vector with its tail at A and its head at G, which you could measure with a measuring stick and a protractor.

Can be extended to three (or more) dimensions

The physics textbook titled College Physics by Mendenhall, Eve, Keys, and Sutton contains the following example.

"If a man climbs up the mast of a ship sailing east while the tide caries it south, then that sailor may have three displacements at right angles, vecAB 30 feet upward, vecBC 100 feet eastward, vecCD 20 feet southward, all with respect to the bed of the ocean. The three displacements vecAB, vecBC, and vecCD are equivalent to the single displacement vecAD; and in the same way, any number of displacements may be added together."

Difficult to draw

Of course, we can't easily draw that 3D vector diagram on a flat sheet of paper or construct it on a flat computer monitor, but we can solve for the displacement vecAD if we are familiar with computations in three dimensions. (An explanation of 3D computations is beyond the scope of this module.)

The parallelogram law

There is a law that states:

The sum of two vectors in a plane is represented by the diagonal of a parallelogram whose adjacent sides represent the two vector quantities.

The resultant

As I mentioned earlier, the sum of two or more vectors is called their resultant .

In general, the resultant is not simply the algebraic or arithmetic sum of the vector magnitudes. Instead, in our original vector equation

vecAC = vecAB + vecBC

vecAC is the resultant of the sum of vecAB and vecBC in the sense that a single vector vecAC would have the same effect as vecAB and vecAD acting jointly.

Geometrical addition

The parallelogram law is a form of geometrical addition. In engineering and physics, it is often used to find graphical solutions to problems involving forces, velocities, displacements, accelerations, electric fields, and other directed quantities. (Directed quantities have both magnitude and direction.)

Solving a vector problem with a parallelogram

Pretend that you walk from point A

Let's use a vector diagram and the parallelogram law to find the resultant displacement vector vecAC.

Image 5 illustrates the use of the parallelogram law to find a graphical solution to this problem.

5

Image 5: Illustration of the parallelogram law.

Put both vector tails at point A

Mentally designate the starting point on your vector diagram as point A. Construct a vector with a length of 5 meters and an angle of 0 degrees relative to the horizontal with its tail at point A. Mentally designate this vector as vecAB.

Construct a second vector with a length of 6 meters at an angle of 30 degrees with its tail at point A. Mentally designate this vector as vecBC.

Parallel lines

Construct a line at least 6 meters long starting at the head of vecBC. Make this line parallel to vecAB.

(There are drawing tools that make it easy to draw a line that is parallel to another line. This is one reason why this is a popular technique for graphically adding vectors.)

Construct another line, at least 7 meters long, starting at the head of vecAB. Make this line parallel to vecBC.

A parallelogram

Note the point where the two lines intersect. The two lines and the two vectors should now form a parallelogram as shown in Image 5 .

Solving for the resultant vector

According to the parallelogram law, the resultant vector, vecAC, is represented by a directed line that extends from point A to the intersection of the two lines that you constructed earlier. In other words, the sum of the two vectors is equal to a diagonal line that extends from the starting point to the opposite corner on the parallelogram that you constructed.

Subtracting vectors

In normal arithmetic subtraction, a subtrahend is subtracted from a minuend to find a difference .

To subtract a subtrahend vector from a minuend vector (in a plane), add 180 degrees to the direction of the subtrahend vector, causing it to point in the opposite direction, and add the modified subtrahend vector to the minuend vector.

If you subtract vecAB in the above example from vecBC, you should be able to show that the difference vector is represented by the other diagonal in the parallelogram. However, you will probably find it easier to do this mathematically rather than doing it graphically.

The horizontal component of the sum of two or more vectors is the sum of the horizontal components of the vectors.

The vertical component of the sum of two or more vectors is the sum of the vertical components of the vectors.

The horizontal and vertical components

The horizontal and vertical components respectively of a vector vecAB are equal to

where

The magnitude of the resultant vector

Given the horizontal and vertical components of the resultant vector, the magnitude of the resultant vector can be found using the Pythagorean theorem as the square root of the sum of the squares of the horizontal and vertical components.

The angle of the resultant vector

The angle that the resultant vector makes with the horizontal axis is the arctangent (corrected for quadrant) of the ratio of the vertical component to the horizontal component.

<!-- File JavaScript01.html -->
<html><body>
<script language="JavaScript1.3">

//The purpose of this function is to receive the adjacent
// and opposite side values for a right triangle and to
// return the angle in degrees in the correct quadrant.
function getAngle(x,y){
  if((x == 0) && (y == 0)){
    //Angle is indeterminate. Just return zero.
    return 0;
  }else if((x == 0) && (y > 0)){
    //Avoid divide by zero denominator.
    return 90;
  }else if((x == 0) && (y < 0)){
    //Avoid divide by zero denominator.
    return -90;
  }else if((x < 0) && (y >= 0)){
    //Correct to second quadrant
    return Math.atan(y/x)*180/Math.PI + 180;
  }else if((x < 0) && (y <= 0)){
    //Correct to third quadrant
    return Math.atan(y/x)*180/Math.PI + 180;
  }else{
    //First and fourth quadrants. No correction required.
    return Math.atan(y/x)*180/Math.PI;
  }//end else
}//end function getAngle

//Test for a range of known angles.
var angIn = 0;
var angOut = 0;
var x = 0;
var y = 0;
while(angIn <= 360){
  x = Math.cos(angIn*Math.PI/180);
  y = Math.sin(angIn*Math.PI/180);
  angOut = getAngle(x,y).toFixed(1);
  document.write("angle = " + angOut + "<br/>");
  angIn = angIn + 30;
}//end while loop

document.write("The End")

</script>
</body></html>

angle = 0.0
angle = 30.0
angle = 60.0
angle = 90.0
angle = 120.0
angle = 150.0
angle = 180.0
angle = 210.0
angle = 240.0
angle = 270.0
angle = -60.0
angle = -30.0
angle = -0.0
The End 

<!-- File JavaScript02.html -->
<html><body>
<script language="JavaScript1.3">

//The purpose of this function is to receive the adjacent
// and opposite side values for a right triangle and to
// return the angle in degrees in the correct quadrant.
function getAngle(x,y){
  if((x == 0) && (y == 0)){
    //Angle is indeterminate. Just return zero.
    return 0;
  }else if((x == 0) && (y > 0)){
    //Avoid divide by zero denominator.
    return 90;
  }else if((x == 0) && (y < 0)){
    //Avoid divide by zero denominator.
    return -90;
  }else if((x < 0) && (y >= 0)){
    //Correct to second quadrant
    return Math.atan(y/x)*180/Math.PI + 180;
  }else if((x < 0) && (y <= 0)){
    //Correct to third quadrant
    return Math.atan(y/x)*180/Math.PI + 180;
  }else{
    //First and fourth quadrants. No correction required.
    return Math.atan(y/x)*180/Math.PI;
  }//end else
}//end function getAngle


//Specify the magnitudes and angles of two vectors.
//Convert the angles from degrees to radians.
var vecABmag = 30;//at an angle of 0
var vecABang = 0*Math.PI/180;//degrees*radians/degrees=radians

var vecBCmag = 40;//at an angle of 135
var vecBCang = 135*Math.PI/180;//degrees*radians/degrees=radians


//Compute the horizontal and vertical components of each vector.
var vecABh = vecABmag*Math.cos(vecABang);
var vecABv = vecABmag*Math.sin(vecABang);

var vecBCh = vecBCmag*Math.cos(vecBCang);
var vecBCv = vecBCmag*Math.sin(vecBCang);

//Compute the sums of the horizontal and vertical components from
// the two vectors to get the horizontal and vertical component
// of the resultant vector.
var vecResultH = vecABh + vecBCh;
var vecResultV = vecABv + vecBCv;

//Use the Pythagorean theorem to compute the magnitude of the
// resultant vector.
var vecResultMag = Math.sqrt(Math.pow(vecResultH,2) + 
                   Math.pow(vecResultV,2));

//Compute the angle of the resultant.
vecResultAng = getAngle(vecResultH,vecResultV);

document.write("Hello from JavaScript" + "<br/>")
document.write("vecABv = " + vecABv + "<br/>");
document.write("vecABh = " + vecABh + "<br/>");
document.write("vecBCv = " + vecBCv + "<br/>");
document.write("vecBCh = " + vecBCh + "<br/>");
document.write("vecResultMag = " + vecResultMag + " <br/>");
document.write("vecResultAng = " + vecResultAng + " <br/>");


document.write("The End")

</script>
</body></html>

Hello from JavaScript
vecABv = 0
vecABh = 30
vecBCv = 28.284271247461902
vecBCh = -28.2842712474619
vecResultMag = 28.336261665087125
vecResultAng = 86.5286817554714
The End 

<!-- File JavaScript03.html -->
<html><body>
<script language="JavaScript1.3">

//The purpose of this function is to receive the adjacent
// and opposite side values for a right triangle and to
// return the angle in degrees in the correct quadrant.
function getAngle(x,y){
  if((x == 0) && (y == 0)){
    //Angle is indeterminate. Just return zero.
    return 0;
  }else if((x == 0) && (y > 0)){
    //Avoid divide by zero denominator.
    return 90;
  }else if((x == 0) && (y < 0)){
    //Avoid divide by zero denominator.
    return -90;
  }else if((x < 0) && (y >= 0)){
    //Correct to second quadrant
    return Math.atan(y/x)*180/Math.PI + 180;
  }else if((x < 0) && (y <= 0)){
    //Correct to third quadrant
    return Math.atan(y/x)*180/Math.PI + 180;
  }else{
    //First and fourth quadrants. No correction required.
    return Math.atan(y/x)*180/Math.PI;
  }//end else
}//end function getAngle


//Specify the magnitudes and angles of four vectors.
//Convert the angles from degrees to radians.
var vecAmag = 12;
var vecAang = 20*Math.PI/180;//degrees*radians/degrees=radians

var vecBmag = 8;
var vecBang = 120*Math.PI/180;//degrees*radians/degrees=radians

var vecCmag = 10.5;
var vecCang = -74*Math.PI/180;//degrees*radians/degrees=radians

var vecDmag = 12.5;
var vecDang = 145*Math.PI/180;//degrees*radians/degrees=radians


//Compute the horizontal and vertical components of each vector.
var vecAh = vecAmag*Math.cos(vecAang);
var vecAv = vecAmag*Math.sin(vecAang);

var vecBh = vecBmag*Math.cos(vecBang);
var vecBv = vecBmag*Math.sin(vecBang);

var vecCh = vecCmag*Math.cos(vecCang);
var vecCv = vecCmag*Math.sin(vecCang);

var vecDh = vecDmag*Math.cos(vecDang);
var vecDv = vecDmag*Math.sin(vecDang);

//Compute the sums of the horizontal and vertical components from
// the four vectors to get the horizontal and vertical component
// of the resultant vector.
var vecResultH = vecAh + vecBh + vecCh + vecDh;
var vecResultV = vecAv + vecBv + vecCv + vecDv;

//Use the Pythagorean theorem to compute the magnitude of the
// resultant vector.
var vecResultMag = Math.sqrt(Math.pow(vecResultH,2) +
                   Math.pow(vecResultV,2));

//Compute the angle of the resultant vector.
vecResultAng = getAngle(vecResultH,vecResultV);

document.write("Hello from JavaScript" + "<br/>")
document.write(
  "vecResultMag = " + vecResultMag.toFixed(2) + " <br/>");
document.write(
  "vecResultAng = " + vecResultAng.toFixed(2) + " <br/>");


document.write("The End")

</script>
</body></html>

Hello from JavaScript
vecResultMag = 8.11
vecResultAng = 90.49
The End