Vectors - Fundamentals and Operations
Vectors and Direction
A study of motion will involve the introduction of a variety of quantities which are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories - vectors and scalars. A vector quantity is a quantity which is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity which is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces which occur in two dimensions.
Examples
of vector quantities which have been previously
discussed include displacement,
velocity, acceleration,
and force. Each of
these quantities are unique in that a full description of
the quantity demands that both a magnitude and a direction
are listed. For example, suppose your teacher tells you "A
bag of gold is located outside the classroom. To find it,
displace yourself 20 meters." This statement may provide
yourself enough information to pique your interest; yet,
there is not enough information included in the statement to
find the bag of gold. The displacement required to find the
bag of gold has not been fully described. On the other hand,
suppose your teacher tells you "A bag of gold is located
outside the classroom. To find it, displace yourself from
the center of the classroom door 20 meters in a direction 30
degrees to the west of north." This statement now provides a
complete description of the displacement vector - it lists
both magnitude (20 meters) and direction (30 degrees to the
west of north) relative to a reference or starting position
(the center of the classroom door). Vector quantities are
not fully described unless both magnitude and direction are
listed.
Vector
quantities are often represented by scaled vector
diagrams. Vector diagrams depict a vector by use of an
arrow drawn to scale in a specific direction. Vector
diagrams were introduced and used in earlier units to depict
the forces acting upon an object. Such diagrams are commonly
called as free-body
diagrams. An example of a scaled vector diagram is shown
in the diagram at the right. The vector diagram depicts a
displacement vector. Observe that there are several
characteristics of this diagram which make it an
appropriately drawn vector diagram.
- a scale is clearly listed
- a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail.
- the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).
Conventions for Describing Directions of Vectors
Vectors
can be directed due East, due West, due South, and due
North. But some vectors are directed northeast (at a
45 degree angle); and some vectors are even directed
northeast, yet more north than east. Thus, there is a
clear need for some form of a convention for identifying the
direction of a vector which is not due East, due
West, due South, or due North. There are a variety of
conventions for describing the direction of any vector. The
two conventions which will be discussed and used in this
unit are described below:
- The direction of a vector is often expressed as an angle of rotation of the vector about its "tail" from either east, west, north, or south. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction) of 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction).
- The direction of a vector is often expressed as an
counterclockwise angle of rotation of the vector about
its "tail" from due East. Using this
convention, a vector with a direction of 30 degrees is a
vector which has been rotated 30 degrees in a
counterclockwise direction relative to due east. A vector
with a direction of 160 degrees is a vector which has
been rotated 160 degrees in a counterclockwise direction
relative to due east. A vector with a direction of 270
degrees is a vector which has been rotated 270 degrees in
a counterclockwise direction relative to due east. This
is one of the most common conventions for the direction
of a vector and will be utilized throughout this
unit.
Two illustrations of the second convention (discussed above) for identifying the direction of a vector are shown below.
Observe in the first example that the
vector is said to have a direction of 40 degrees. You can
think of this direction as follows: suppose a vector
pointing East had its tail pinned down and
then the vector was rotated an angle of 40 degrees in the
clockwise direction. Observe in the second example that the
vector is said to have a direction of 240 degrees. This
means that the tail of the vector was pinned down and the
vector was rotated an angle of 240 degrees in the
counterclockwise direction beginning from due east. A
rotation of 240 degrees is equivalent to rotating the vector
through two quadrants (180 degrees) and then an additional
60 degrees into the third quadrant.
Representing
the Magnitude of a Vector
The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale. For example, the diagram at the right shows a vector with a magnitude of 20 miles. Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector arrow is drawn with a length of 4 cm. That is, 4 cm x (5 miles/1 cm) = 20 miles.
Using the same scale (1 cm = 5 miles), a displacement vector which is 15 miles will be represented by a vector arrow which is 3 cm in length. Similarly, a 25 mile displacement vector is represented by a 5-cm long vector arrow. And finally, an 18 mile displacement vector is represented by a 3.6-cm long arrow. See the examples shown below.
In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated (such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction. Directions are described by the use of some convention. The most common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.
In the remainder of this lesson, in the entire unit, and in future units, scaled vector diagrams and the above convention for the direction of a vector will be frequently used to describe motion and solve problems concerning motion. For this reason, it is critical that you have a comfortable understanding of the means of representing and describing vector quantities. Some practice problems are available on-line at the following WWW page: