Name: _______________________________ Period : ________ Date :_____________
Vectors and Direction
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.
Page 1 of 5
Name: _______________________________ Period : ________ Date :_____________
Vectors and Direction
There are a variety of methods for determining the magnitude and direction of the result of adding two or more
vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are:


the Pythagorean theorem and trigonometric methods
the head-to-tail method using a scaled vector diagram
The Pythagorean Theorem
The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors
that make a right angle to each other. The method is not applicable for adding more than two vectors or for
adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation
that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.
To see how the method works, consider the following problem:
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 that 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
Page 2 of 5
Name: _______________________________ Period : ________ Date :_____________
Vectors and Direction
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.
Questions:
1) 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.
Page 3 of 5
Name: _______________________________ Period : ________ Date :_____________
Vectors and Direction
2) The resulting vector from this situation would most likely be?
5
+
3
3) Find the value of the resultant vector in the diagram shown.
60 m
80 m
4) A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North.
What was the distance traveled and the displacement of the Physics teacher? (hint: make a sketch of the
vectors)
Page 4 of 5
Name: _______________________________ Period : ________ Date :_____________
Vectors Addition
Page 5 of 5