HPP A9
North by Northwest - An Introduction to Vectors
Exploration
GE 1.
Let's suppose you and a friend are standing in the parking lot near the Science
Building. Your friend says, "I am going to run at a speed of 5 [m/s]. Guess
where I will be in 10 [s]?"
1. Would you be able to answer your friend's question? Are you missing an
important piece of information?
Invention
Quantities that have direction, as well as magnitude, are called vectors and can be represented as
arrows where the length of the arrow represents the magnitude and the arrow gives the
direction.

Quantities that do not have direction are called scalars.
Application
GE 2.
1. Which of the following are vectors and which are scalars?
a) Wind speed 15mph
b) Wind 15mph NE
c) Calories in a donut
d) Push
e) Temperature
f) Gradient of a hill
g) Where the science building is, relative to where you live.
h) How far away the science building is from where you live.
2. Would velocity for one-dimensional motion be considered a scalar or
vector?
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 2010 The Humanized Physics Project
Supported in part by NSF-CCLI Program under grants DUE #00-88712 and DUE #00-88780
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3. How about acceleration?
Invention: How do you add vectors?
For one dimension
Graphically: tail-to-head
+
3 units
=
=
5 units
3 units + 5 units
8 unit s
The sum of vectors is called the resultant vector (resultant for short).
resultant has a magnitude of 8 units and it is pointing to the right.
In the case above the
As on number line: Vector going one direction (say to the right) is taken to be on number line
pointing in positive direction. Vector going in opposite direction is considered to be pointing in
negative direction.
Adding vectors going in same direction is like adding positive numbers.
Adding vectors going in opposite directions is like adding positive and negative numbers i.e.
subtraction
+
3 units
+
(- 5 units)
=
=
3 units – 5 units
=
=
(- 2 units)
Like adding or subtracting regular numbers – the order does not matter.
Application
GE 3.
1. Draw the resultant vector of the following. Give its magnitude and
direction.
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+
8 units
+
4 units
=
5 units
Invention - Adding vectors in two dimensions
Vectors in the plane can be added in a similar fashion to the linear case shown above. Suppose
we want to add vectors A and B shown below.
B
A
We can add them graphically be taking B and moving it over to A, placing the tail of B at the tip
of A. The vector sum is drawn from the tail of A to the tip of B.
A
B
A+B
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Application
GE 4.
1. Add the displacement vectors A and B graphically.
B
A
2. What is the magnitude of the vector sum?
3. What is the direction of the vector sum, measured from the horizontal?
Exploration
The figure below shows a displacement vector for a person who goes from the origin to the
coordinates (4 [m], 5 [m]).
Y
X
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GE 5.
1. Is walking directly from (0,0) to (4,5) the only path a person could take?
2. If the person takes an indirect path from (0,0) to (4,5), is the displacement
vector the same?
Suppose there is a large pile of boxes in the direct path as shown below.
Y
X
3. What would be a simple way to bypass the boxes and arrive at the same
destination?
Invention
Any vector in the plane can be visualized as the addition of a vector in the x direction and a
vector in the y direction.
A = Ax + Ay
The figure below shows this vector addition graphically.
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Y
A = Ax + Ay
Ay
X
Ax
The vector in the x direction, Ax, is called the x-component. The vector in the y direction, Ay, is
called the y-component. The magnitude of the components can be obtained from the original
vector by using trigonometry.
Y
q
X
Ax  A cos(q ) and Ay  A sin( q )
Application
GE 6.
1. Find the x and y components of vector A shown below. It has a length of
5.0 [m/s].
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55º
Ax =
Ay =
Invention
If you know the components you can use trigonometry to find the magnitude of the vector and
the angle it makes.
 Ay 

 Ax 
It is always a good idea to draw a picture pointing out which angle you calculate, to avoid any
ambiguity.
A  Ax2  Ay2
q  tan 1 
Invention
Vector addition can be done very precisely using components. If you know the components of
two vectors A and B, then the vector sum C is determined by getting its components as shown
below.
Cx = Ax + Bx
Cy = Ay + By
Application
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B
A
GE 7.
1. Find the x and y components of vector A and B in the above figure. You
can use a ruler to measure lengths.
Ax =
Ay =
Bx =
By =
2. Calculate the vector sum C of A and B using components.
Cx =
Cy =
3. Calculate the magnitude of C using the components.
4. Calculate the angle that C makes with the positive x-direction, using
components.
5. Compare your magnitude and direction of C with what you obtained
graphically in GE 4.
Activity Guide
 2010 The Humanized Physics Project