Lesson 1: Vectors - Fundamentals and Operations
Lesson 2: Projectile Motion
Describing Projectiles with Numbers:
Lesson 3 : Forces in Two Dimensions
Lesson 1: Vectors - Fundamentals and Operations
A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). This process of adding two or more vectors has already been discussed in an earlier unit. Recall in our discussion of Newton's laws of motion, that the net force experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object. That is the net force was the result (or resultant) of adding up all the force vectors. During that unit, the rules for summing vectors (such as force vectors) were kept relatively simple. Observe the following summations of two force vectors:
These rules for summing vectors were applied to free-body diagrams in order to determine the net force (i.e., the vector sum of all the individual forces). Sample applications are shown in the diagram below.
In this unit, the task of summing vectors will be extended to more complicated cases in which the vectors are directed in directions other than purely vertical and horizontal directions. For example, a vector directed up and to the right will be added to a vector directed up and to the left. The vector sum will be determined for the more complicated cases shown in the diagrams below.
There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods which will be discussed in this lesson and used throughout the entire unit are:
The Pythagorean Theorem
The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors which make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors which are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation which relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.
Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.
This problem asks to determine the result of adding two displacement vectors which are at right angles to each other. The result (or resultant) of walking 11 km north and 11 km east is a vector directed northeast as shown in the diagram to the right. Since the northward displacement and the eastward displacement are at right angles to each other, the Pythagorean theorem can be used to determine the resultant (i.e., the hypotenuse of the right triangle).
The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6 km. Later, the method of determining the direction of the vector will be discussed.
Let's test your understanding with the following two practice problems. In each case, use the Pythagorean theorem to determine the magnitude of the vector sum. When finished, click the button to view the answer.
Using Trigonometry to Determine a Vector's Direction
The direction of a resultant vector can often be determined by use of trigonometric functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA from their course in trigonometry. SOH CAH TOA is a mnemonic which helps one remember the meaning of the three common trigonometric functions - sine, cosine, and tangent functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The tangent function relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The three equations below summarize these three functions in equation form.
These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement. The process begins by the selection of one of the two angles (other than the right angle) of the triangle. Once the angle is selected, any of the three functions can be used to find the measure of the angle. Write the function and proceed with the proper algebraic steps to solve for the measure of the angle. The work is shown below.
Once the measure of the angle is determined, the direction of the vector can be found. In this case the vector makes an angle of 45 degrees with due East. Thus, the direction of this vector is written as 45 degrees. (Recall from earlier in this lesson that the direction of a vector is the counterclockwise angle of rotation which the vector makes with due East.)
The measure of an angle as determined through use of SOH CAH TOA is not always the direction of the vector. The following vector addition diagram is an example of such a situation. Observe that the angle within the triangle is determined to be 26.6 degrees using SOH CAH TOA. This angle is the southward angle of rotation which the vector R makes with respect to West. Yet the direction of the vector as expressed with the CCW (counterclockwise from East) convention is 206.6 degrees.
Test your understanding of the use of SOH CAH TOA to determine the vector direction by trying the following two practice problems. In each case, use SOH CAH TOA to determine the direction of the resultant. When finished, click the button to view the answer.
In the above problems, the magnitude and direction of the sum of two vectors is determined using the Pythagorean theorem and trigonometric methods (SOH CAH TOA). The procedure is restricted to the addition of two vectors which make right angles to each other. When the two vectors which are to be added do not make right angles to one another, or when there are more than two vectors to add together, we will employ a method known as the head-to-tail vector addition method. This method is described below.
The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the head-to-tail method is employed to determine the vector sum or resultant. A common Physics lab involves a vector walk. Either using centimeter-sized displacements upon a map or meter-sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position. Suppose that you were given a map of your local area and a set of 18 directions to follow. Starting at home base, these 18 displacement vectors could be added together in consecutive fashion to determine the result of adding the set of 18 directions. Perhaps the first vector is measured 5 cm, East. Where this measurement ended, the next measurement would begin. The process would be repeated for all 18 directions. Each time one measurement ended, the next measurement would begin. In essence, you would be using the head-to-tail method of vector addition.
The head-to-tail method involves
drawing a vector to scale
on a sheet of paper beginning at a designated starting
position. Where the head of this first vector ends, the tail
of the second vector begins (thus, head-to-tail
method). The process is repeated for all vectors which are
being added. Once all the vectors have been added
head-to-tail, the resultant is then drawn from the tail of
the first vector to the head of the last vector; i.e., from
start to finish. Once the resultant is drawn, its length can
be measured and converted to real units using the
given scale. The direction of
the resultant can be determined by using a protractor and
measuring its counterclockwise angle of rotation from due
© Tom Henderson