VECTORS: RESULTANTS AND COMPONENTS
It is left as an exercise for the student to show that vector addition is commutative, using the
above example. Specifically, make a scale drawing showing that traveling 3 miles north and
then 4 miles east yields the same resultant as above.
It is also reasonably obvious that more than two vectors can be added. One can travel three
miles east and then three miles north and then three miles west and arrive at a point three miles
north of the starting point. The sum of these three displacements is a resultant displacement
of three miles north. (If this is not immediately apparent, sketch it.)
A student problem is to find the net or resultant displacement if a person travels 9 miles south
and then 12 miles east and then 25 miles north. Make a scale drawing and determine the
magnitude and direction of the resultant displacement. A scale of 2 miles per centimeter or
4 miles per inch will fit the drawing on standard paper.
About 20 miles at 53 north of east.
Components of a vector are vectors, which when added, yield the vector. For example, as
shown in the previous section (Figure 10), traveling 3 miles north and then 4 miles east yields
a resultant displacement of 5 miles, 37 north of east. This example demonstrates that
component vectors of any two non-parallel directions can be obtained for any resultant vector
in the same plane. For the purposes of this manual, we restrict our discussions to two
dimensional space. The student should realize that vectors can and do exist in three dimensional
One could write an alternate problem: "If I am 5 miles from where I started northeast along a
line 37 N of east, how far north and how far east am I from my original position?" Drawing this
on a scale drawing, the vector components in the east and north directions can be measured to
be about 4 miles east and 3 miles north. These two vectors are the components of the resultant
vector of 5 miles, 37 north of east.
Component vectors can be determined by plotting them on a rectangular coordinate system. For
example, a resultant vector of 5 units at 53 can be broken down into its respective x and y
magnitudes. The x value of 3 and the y value of 4 can be determined using trigonometry or
graphically. Their magnitudes and position can be expressed by one of several conventions
including: (3,4), (x=3, y=4), (3 at 0 , 4 at 90 ), and (5 at 53 ). In the first expression, the first
term is the x-component (F ), and the second term is the y-component (F ) of the associated