Doing Physics—Using Scalars and Vectors
Scalars
Many physical quantities can be described completely by a number and a unit.
For instance, the mass of an object might be 6 kg, and its temperature 30 C.
When a physical quantity is described by a single number, we call it a scalar quantity.
Some scalar quantities: mass, temperature, time, density, electric charge, speed
A scalar can be positive, negative, or zero.
Examples: 0 seconds,
4 kg,
15 inches,
a volume of 2 gal.
Vectors
While many quantities that we meet in physics are scalars, there are others that are not.
For these quantities, a single number tells only part of the story.
For the scalars that we gave above, we didn’t need to specify any direction. This isn’t always
the case.
A quantity that deals inherently with direction is called a vector quantity.
Some vector quantities: velocity, force, electric field, acceleration, displacement
Vectors
To fully describe the motion of an airplane, we have to say not only how fast it is moving, but
also in what direction.
And to describe a force, in addition to giving its magnitude we need to specify the direction
in which the force acts.
These examples, velocity and force, show that vectors quantities have both magnitude
and direction.
The magnitude of a vector gives us the size, or the “how much.”
Because size can never be negative, the magnitude of a vector cannot be negative.
The direction of a vector tells us the orientation of the quantity; it tells us the way the
quantity points in space.
The direction might be specified as an angle, like 30˚, or by something like “northwest”, or
by saying “to the left.”
A vector quantity is not complete until both its magnitude and direction are specified.
Conceptual Check
There are places where the temperature can be +20 C at one time, and -20 C
at another time.
Does the sign of the temperature indicate a direction?
Is temperature a vector quantity?
Displacement Vectors
The first vector quantity we need to talk about is displacement.
Displacement is simply the change in position of an object.
A displacement vector starts at an object’s initial
position and ends at its final position. It doesn’t matter
what the object did in between these two positions.
A displacement vector is always a straight line, no
matter how curved or squiggly the actual path of
motion is. (All vectors are straight lines)
This means that displacement is not the same thing as
distance.
If one travels for a great distance, but comes back to
where they started, then their displacement is zero!
What does a displacement vector tell us, if not the
distance?
Its magnitude tells us how far you ended up from
where you started (the “straight-line” distance).
Its direction tells us which way you moved overall.
Representing Vectors
The previous example shows that a vectors can be conveniently represented as an arrow.
This is the case since an arrow has a magnitude (how long the arrow is) as well as a
direction (the way the arrow points).
This graphical or pictorial representation will come in handy later on.
The velocity of this car is 100 m/s (magnitude)
to the left (direction).
This boy pushes on his friend with a force of
25 N (magnitude)to the right (direction).
Note that while arrows have a magnitude and direction, they don’t have a particular
location. That is, vectors have no home. A vector can be moved about and still be the same
vector, as long as its length and direction don’t change.
So this vector
is the same as this vector
Two vectors are equal when they have
exactly the same magnitude and exactly the same direction.
Though vectors have both a magnitude and a direction, they do not have a “home.”
That is, we can slide them around as long as we don’t change the size or direction of
the vector.
Vectors have no home!
Although they have the same magnitude
(or length), these two vectors are not
equal, since they point in different
directions.
Although they have the same direction,
these two vectors are not equal, since they
have different magnitudes.
Two vectors are equal when they have
exactly the same magnitude and exactly the same direction.
Vector Notation
Because vectors are different from ordinary numbers (scalars), we label them in a special way.
Labelling Vectors
In the textbook, vectors are given a letter name in bold and with an arrow over them.
In your hand-written work, indicate a vector with an arrow over the name of the vector.
A
Notice that it is not

A
The arrow above the letter always points to the right,
regardless of the direction of the vector.
Important: When you write a symbol for a vector, always put an arrow over it. If you don’t
put an arrow, than the symbol represents a scalar, and vectors and scalars are very different!
⟶
Thus r and A are symbols for vectors, whereas,
symbols for scalars.
⟶
In fact,
r
and
A
r
⟶
and
⟶
A, without the arrows, are
are how we denote the magnitudes of the vectors.
Magnitude of
⟶
A
=
⟶
|A|
=
A
Vector Math
Calculations with scalar quantities use ordinary arithmetic.
For example, 5 kg + 8 kg = 13 kg
or
9 s – 5 s = 4 s.
Vector calculations, on the other hand, do not use ordinary arithmetic.
Consider, for example, our friend Sam, who walks along 12th Street and then goes down the
bike path:
The final result is the same as though Sam had
started at his initial position and undergone a
single displacement, as shown.
Since the two individual displacements, if taken
together, result in the same single displacement,
we write the vector equality as shown.
We call the single vector the vector sum or the
resultant or the net vector.
Notice that the magnitude of the vector sum
is not equal to the sum of the individual
magnitudes.
Question: In what situation is the magnitude of the vector sum equal to the sum of the
individual magnitudes?
Adding Vectors
We can use the fact that vectors have no home to help us add vectors.
(2) Then draw the resultant
vector⟶
⟶
from the tail of D to the tip of E.
(1) Slide the tail of⟶the
second vector, E, to the
⟶
tip of the first vector, D.
This is the tail-to-tip method.
Adding Vectors
A second way to add vectors is the parallelogram method.
(1) Put the vectors tail to tail.
(2) With dashed lines, mark the parallelogram.
(3) The resultant vector then points from the
tails to the opposite corner, along the diagonal.
This is the parallelogram method.
Scalar Multiplication
Recall that a scalar is a single number (often with some unit), which can positive, zero, or
negative.
Question: What might you guess happens when we multiply a vector by a scalar?
We can change the length (the magnitude) of a vector by multiplying it by a scalar.
We generally use lower case letters for scalars: a, b, c, and so on.
Note that we cannot change the direction of a vector with scalar multiplication.
The exception to this is when multiplying by a negative scalar, as we’ll see.
(To change the direction of a vector in a general way, we need matrix multiplication.)
Multiplication by a positive scalar:
(1) If the scalar is greater than one, scalar multiplication will
stretch a vector, making it longer
(2) If the scalar is less than one (but positive), scalar
multiplication will shrink the vector, making it shorter.
A
2A
1
2
A
Multiplication by a negative scalar:
(1) If the scalar is negative and its absolute value is greater than
one, scalar multiplication will flip and stretch a vector.
(2) If the scalar is negative and its absolute value is less than
one, scalar multiplication will flip and shrink a vector.
A
2 A
 12 A
The definition of multiplication by a negative scalar provides the basis for
defining vector subtraction.
⟶
⟶
Consider the vector subtraction A – B.
⟶
⟶
To carry this out, we just add A and (– B).
⟶
⟶
⟶
⟶
A – B = A + (–B).
Subtracting Vectors Graphically
Vector worksheet
Vector addition and multiplication by a scalar
Coordinate Systems and
Vector Components
So far we’ve worked with vectors graphically, using pictures to
show scalar multiplication and vector addition and subtraction.
The graphical approach is helpful in getting a qualitative sense of
the motion—the direction of the acceleration, for instance—but
it isn’t especially good for an exact description of an object’s
motion.
The graphical approach
We now want to develop some ideas which will allow us to work
with vectors in a more quantitative way, so that we can do some
real calculations with real numbers.
To this end, we lay out here the basics of coordinate systems and
components.
We will usually use Cartesian
coordinates, the familiar rectangular
grid with perpendicular axes.
As usual, we label the positive end of
each axis.
Component Vectors
We can resolve (or decompose) any
vector
into
two
perpendicular
component vectors.
One of these component vectors is
parallel to the x-axis, and we label it
Ax
The other component vector is
parallel to the y-axis, and we label it
Ay
Notice that the vector A is the vector sum of Ax and Ay .
A  Ax  Ay
⟶
The vector Ax is called the x-component vector.
⟶
⟶
Ax is the projection of A along the x-axis.
⟶
The vector Ay is called the y-component vector.
⟶
⟶
Ay is the projection of A along the y-axis.
Components
⟶
Suppose we have⟶ a vector
A that has been resolved into
⟶
component vectors Ax and Ay parallel to the coordinate axes.
We can describe each component vector with a single number
(a scalar) called the component (which is different from a component vector!).
⟶
⟶
Ax = The x-component of the vector A. | Ax | is the magnitude of Ax
Ay = The y-component of the vector A. | Ay | is the magnitude of Ay.
⟶
⟶
The components Ax and Ay tell us how long the component vectors are,
and in which direction they point.
The direction is captured in the sign (+ or –) of the components.
Note
Ax and Ayare component vectors; they have a
magnitude (which is always positive) and a direction.
Ax and Ay are components. They are numbers
(with units) that can be positive or negative.
Ax and Ay are not the magnitudes of Ax and Ay.
| Ax| is the magnitude of Ax . |Ay| is the magnitude of Ay .
Vector worksheet
Resolving vectors into components
opposite
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
sin  
To remember this, we can use the helpful mnemonic soh-cah-toa:
soh-cah-toa
o(pposite)
sin  
h(ypotenuse)
a(djacent )
cos  
h( ypotenuse)
o( pposite)
tan  
a (djacent )
Decomposing a vector into its x- and y-components.
The y-component of A
The x-component of A
Note
When it comes to figuring out the angle and the signs of the
components, always draw a picture to guide you!
Each decomposition requires that you pay close attention to the
direction in which the vector points and the angles that are defined.
It’s generally best to use angles between 0 and 90 (for which sine and
cosine are non-negative) and to then put in a negative sign “by hand.”
Draw a picture with coordinates, label the angle, label the components.
Calculate and check!
⟶
Example. Resolve A into its x- and y-components.
⟶
A, A = 14
Ay will be positive
y
 = 40
140
Ax will be negative
x
Answer. It’s up to us which angle we choose. Let’s work with
the angle between the vector and the negative x-axis. Call it .
⟶
We’re given that the magnitude of A is 14.
Ax   A cos   14 cos 40  14(.766)  10.72
Ay   A sin   14sin 40  14(.643)  8.99
Notice that we put in the negative sign “by hand” for Ax since we knew from
the picture that it would be negative. How could we check our values?
Example.
Find the x- and y-components of
the acceleration vector shown here.
Prepare: Draw a picture, labeling the components and angle.
On the picture, note the signs of the components.
Solve: Both the x- and y-components are negative, so we have
ax  a cos 30  (6.0 m/s 2 ) cos 30  5.2 m/s 2
a y  a sin 30  (6.0 m/s 2 )sin 30  3.0 m/s 2
Asses: The units of the components ax and ay are the same as the
units of the acceleration vector. Notice that our picture guided us in
knowing what signs to put in for the calculations.
Checking Understanding

Ax is the __________ of the vector A.
A.
B.
C.
D.
E.
magnitude
x-component
direction
size
displacement
Answer

Ax is the __________ of the vector A.
B. x-component
Checking Understanding
Which of the vectors
below best represents the

vector sum P + Q?
Slide 3-13
Answer
Which of the vectors below best represents the
 
vector sum P + Q?
Slide 3-14
Checking Understanding
Which of the vectors below best represents the
 
difference P – Q ?
Answer
Which of the vectors below best represents the
 
difference P – Q ?
Slide 3-16
Checking Understanding
Which of the vectors below best represents the
 
difference Q – P?
A.
B.
C.
D.
Answer
Which of the vectors below best represents the
 
difference Q – P ?
B.
What are the x- and y-components of this vector?
A.
B.
C.
D.
E.
3, 2
2, 3
-3, 2
2, -3
-3, -2
What are the x- and y-components of this vector?
B. 2, 3
What are the x- and y-components of this vector?
A.
B.
C.
D.
E.
3, 4
4, 3
-3, 4
4, -3
3, -4
What are the x- and y-components of these vectors?
E. 3, -4
The following vector has length 4.0 units.
What are the x- and y-components of this vector?
A.
B.
C.
D.
E.
3.5, 2.0
-2.0, 3.5
-3.5, 2.0
2.0, -3.5
-3.5, -2.0
The following vector has length 4.0 units.
What are the x- and y-components of this vector?
B. -2.0, 3.5
The following vector has length 4.0 units.
What are the x- and y-components of this vector?
A.
B.
C.
D.
E.
3.5, 2.0
2.0, 3.5
-3.5, 2.0
2.0, -3.5
-3.5, -2.0
The following vector has length 4.0 units.
What are the x- and y-components of this vector?
E. -3.5, -2.0
Show that the x- and y-components of a vector don’t
depend on the angle we use to describe its direction.
y
Ax
Ay
20
70
Ax
x
Ay
A, magnitude = 8 units
Using 20
Using 70
Ax = 8 cos(20) = 7.52
Ax = 8 sin(70) = 7.52
Ay = - 8 sin(20) = - 2.74
Ay = - 8 cos(70) = - 2.74
Notice that we could also use the “standard position” angle.
In this case, we don’t need to put the negative sign in by hand.
This is because the sine and cosine functions will take care of
the signs for us.
y
340
x
A, magnitude = 8 units
Ax = 8 cos(340) = 7.52
Ay = 8 sin(340) = - 2.74
Vector worksheet
Using trigonometry with vectors
Working with components
We’ve seen how to add vectors graphically, but there’s an easier
and more accurate way . . .
using components!
Consider the vectors shown below.
B
B
By
The component vectors for B are then:
A
Bx
A
Ay
The component vectors for A are then:
Ax
Let’s look at the vector sum C = A + B below:
B
B
A
A
C=A+B
C=A+B
Cx
B
Ay
B
By
Bx
A
Ax
By
A
Ay
Ax
Bx
Cy
We can see that the component vectors of C are the sum of the
component vectors of A and B.
The same is true of the components: Cx = Ax + Bx and Cy = Ay + By.
Adding vectors using components
In general, if D = A + B +⟶ C + . . . , then the x- and y-components
of the resultant vector D are
⟶
⟶
⟶
⟶
Dx = Ax + Bx + Cx + . . .
Dy = Ay + By + Cy + . . .
In words: To add vectors, add their like components.
This method of vector addition is an algebraic approach, rather than
our previous graphical methods.
Example.
Suppose C = A + B.
Vector A has components Ax = 5, Ay = 6 and
vector B has components Bx = 4, By = 2.
(a) What are the x- and y-components of vector C ?
(b) Draw a coordinate system and on it show vectors A, B, and C.
Answers:
(a) Cx = Ax + Bx = 5 + 4 = 9
Cy = Ay + By = 6 + 2 = 8
y
(b)
B
A
C
x
Example. A bird flies from 100 m due east from a tree, then 200 m
northwest (that is, 45 west of north). What is the bird’s net
displacement?
Answer:
Prepare. Let’s draw a picture with the bird starting out at the
origin. We’ll call the first flight vector A, and the second flight
vector B. Then the total displacement will be C = A + B.
(example continued)
It may be helpful to redraw this picture with all of the
tails at the origin, as shown below:
Solve. To add the vectors
algebraically, we need their
components. From the picture
we find:
Ax = 100 m
Bx = -(200 m) cos 45 = - 141 m
Ay = 0 m
By = (200 m) sin 45 = 141 m
Put in negative sign by hand
Cx = Ax + Bx = 100 m + (-141 m) = - 41 m
Cy = Ay + By = 0 m + 141 m = 141 m
 Cy 
1  141 m 
  tan 
  tan 
  74
C
41
m


 x 
C  Cx 2  C y 2  (41 m)2  (141 m)2  147 m
1
Vector worksheet
Adding vectors by components