Introduction to Vectors
On this web page, you will have an opportunity to
investigate the properties of vectors. This exploration will reinforce
the concepts in Lesson 12.5 of Discovering Advanced Algebra: An
Investigative Approach.
Sketch
A vector can be thought of as a
directed line segment. Examples of vectors are distance, velocity, and
acceleration. A vector has a numerical length, or magnitude,
and a direction.
Vectors are usually represented by segments with
arrowheads on one end. The end with the arrowhead is called the
vector's head or tip, and the other
end is the tail. This sketch shows six different
vectors. You can move vectors a, b,
d, e, and f
around by dragging their tails, which are shown as red dots. The vector
that is completely red can be changed by dragging either its head or
its tail.
Investigate
- There's a way to represent vectors other than as
arrows. You can move the vector to put its tail at the origin and then
describe it by giving the x- and y-coordinates of
its head. This representation is is called the rectangular
form of the vector. Move vector in the
sketch to see that its rectangular form is
 .
Then move the other vectors so that
you can give their rectangular forms.
- To add two vectors, move them so that the tip of the
first vector is at the tail of the second vector; the sum or resultant
is the vector from the tail of the first vector to the tip of the
second vector. Form the resultant vector a + b
by placing the tail of b at the tip of a.
What is the rectangular form of the resultant?
- Repeat this process to find the rectangular forms of
these resultants:
i. b + a
ii. d + e
iii. b + f
iv. a + e
- How do a + b and b
+ a compare?
- Find a formula for vector addition of vectors in
rectangular form and complete this statement:
If a =  and b
=  , then the sum a
+ b is  .
- Subtracting a number is the same as adding its
opposite. It's the same for vectors. The opposite of
vector b is called –b. It has the
same magnitude as b, but it points in the opposite
direction. To find the difference a – b,
you add a + –b. Draw the vector –b.
What is the rectangular form of –b? On your paper,
draw a representation of each difference.
i. b – a
ii. d – e
iii. b – f
iv. a – e
- Now write the rectangular forms of the differences
you found in question 6.
Sketch
This sketch will help you understand an application of
vectors to the flight of an airplane. Vectors can be used to represent
the engine velocity of the plane and the wind velocity. In the sketch
below, you can drag point P at the head of the plane to
change the direction of the plane or the point at its tail to change
the plane's location.
Investigate
- Press Show Wind
Velocity to show the wind velocity vector. You can drag point W
to change the direction or the magnitude of the wind velocity.
- Press Show Engine
Velocity to show the engine velocity vector. You can drag point E
to change the magnitude of the engine velocity.
- Estimate what will happen when these two vectors are
combined. Your answer will show the velocity of the airplane with
respect to the ground.
- Now press Show
Resultant to show the resultant or sum of the wind velocity
vector and the engine velocity vector. Does it match your estimate?
- The sketch uses what geometric shape to combine
vectors? What part of the shape represents the resultant vector?
Summarize your findings with a rule: To find the resultant of two
vectors, make a ___________ with the two vectors as ___________. The
resultant vector is the ___________ of the ___________.
- Describe the resultant when the engine thrust is 300
mi/h and
a. In the same direction as a 60 mi/h wind?
b. In the opposite direction of a 60 mi/h wind?
c. Perpendicular to a 60 mi/h wind?
- The plane needs to fly north at 300 mi/h, but the
wind velocity is 60 mi/h west to east. The pilot should set the
controls for approximately what direction and air speed to get the
desired result? (You may need to use a protractor to estimate the
direction.)
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