Notes: Vectors
Goal: The students will be introduced to the fundamental principles of vector
mathematics.
Objectives: Upon completion of this unit, the students should be able to:
1. Clearly demonstrate an understanding of the difference between a scalar and a
vector quantity.
A scalar is a quantity that can be completely specified by its magnitude with
appropriate units. A scalar has magnitude but no direction. Examples of scalar
quantities are distance, speed, mass, volume and time.
A vector is a physical quantity that has both direction and magnitude. Examples of
vector quantities are displacement, velocity and acceleration. Arrows are often used
to represent vector quantities. The length of the arrow represents the magnitude of
the vector quantity and the direction of the arrow shows the direction of the vector.
The textbook uses boldface type to indicate vector quantities. Scalar quantities are
designated by the use of italics. Thus the speed of a bird would be v = 3.5 m/s but
the velocity would be v = 3.5 m/s, 30° north of east. Handwritten, a vector can be
symbolized by showing an arrow drawn above the abbreviation for a quantity, such
as v = 3.5 m/s, 30° north of east.
Concurrent vectors act simultaneously at the same point. Vectors acting along the
same line are collinear.
2. Add and subtract vectors using the graphical method, and recognize that the order
of vector addition does not matter.
The sum of any two vectors can be found graphically. There are two methods used
to accomplish this: head-to-tail and parallelogram. Regardless of the method used
or the order that the vectors are added, the sum is the same.
Head-to-tail method – the tail of one vector is placed at the head of the other
vector. Neither the direction nor length of either vector is changed. A third vector is
drawn connecting the tail of the first vector to the head of the second vector. This
third vector is called the resultant vector. To find the magnitude of the resultant,
measure its length and direction.
Parallelogram method – it is commonly used when you have concurrent vectors.
The original vectors make the adjacent sides of a parallelogram. A diagonal drawn
from their juncture is the resultant. Its magnitude and direction can be measured.
One way to find the magnitude and direction of the resultant is to draw the situation
to scale on paper. Use a reasonable scale and draw each vector, giving the proper
direction. The magnitude of the resultant vector can then be determined by using a
ruler to measure the length of the resultant vector. The direction of the resultant
vector may be determined by using a protractor to measure the angle between the
first vector and the resultant.
Sample Problem
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Find the resultant of a 3.0-km displacement and a 6.0-km displacement when the
angle between them is 45o.
Note: When using graphical solutions you must decide on a scale for example 1 cm
= 1 km, then to find the resultant you must measure its length and its angle.
Original Vectors
3.0 km at 45°
6.0 km at 0°
Head-to-Tail Method
3.0 km at
Resultant 8.4 km at 15°
45°
3.0 km
45°
15°
45°
6 km at 0°
Parallelogram Method
3.0 km at 45°
Resultant 8.4 kg at 15°
3.0 km
15o
6.0 km at 0°
Sample Problem
A hiker walks 2 km to the North, 3 km to the West, 4 km to the South, 5 km to the
East, 1 more km to the South, and finally 2 km to the West. How far did he end up
from where he started? Hint: What is his resultant?
Vector subtraction is a special case of vector addition:
A – B = A + (-B)
That is, to subtract B from A, a negative B is added to A. The vector –B has the
same magnitude as the vector B, but is in the opposite direction.
-B
A
_
B
=
A
+
-B
=
A-B
A
3. Multiply and divide vectors by scalars.
It is often necessary to multiply a vector quantity by a scalar quantity. For example,
the displacement ∆d of an object moving at a constant velocity v for a time ∆t is
given by the product of the vector v and the scalar ∆t:
∆d = v∆t
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In general, the product of a scalar k and a vector A is a vector whose magnitude is
kA, whose units are the product of the units of k and A, and whose direction is that
of the vector A. Division of vector A by scalar k is treated as if it were multiplied by
the scalar 1/k.
4. Describe situations in terms of frame of reference.
Observers using different frames of reference may measure different
displacements or velocities for an object in motion. If you are moving at 80 km/h
north and a car passes you going 90 km/h, to you the faster car seems to be
moving north at 10 km/h. Someone standing on the side of the road would measure
the velocity of the faster car as 90 km/h toward the north.
This is a simple problem, but sometimes a more systematic method of solving such
problems is beneficial. Write down all the information that is given and that you
want to know in the form of velocities with subscripts.
vse = +80 km/h north (subscript se means the velocity of the slower
car with respect to the earth)
vfe = +90 km/h north (subscript fe means the velocity of the faster car
with respect to the earth)
If you want to know vfs, which is the velocity of the fast car with respect to the
slower car, you write an equation for vfs in terms of the other velocities. On the right
side of the equation the subscripts start with f and end with s. Also, each velocity
subscript starts with the letter that ended the preceding velocity subscript.
vfs = vfe + ves
If we take north to be the positive direction, ves = -vse, and the problem becomes
vfs = +90 km/h – 80 km/h = +10 km/h
The positive sign means that the fast car appears (to the occupants of the slower
car) to be moving north at 10 km/h.
5. Solve problems involving relative velocity.
Vector addition may be used to solve problems involving relative velocities.
Sample Problem
As an airplane taxies on the runway with a speed of 16.5 m/s, a flight attendant
walks toward the tail of the plane with a speed of 1.22 m/s. What is the flight
attendant’s speed relative to the ground?
6. Recognize the independence of perpendicular vector quantities.
Perpendicular vector components are independent of each other.
The graphical method of adding vectors did not require that you decide on a
coordinate system. The sum, or the difference, of vectors is the same no matter
what coordinate system is used. Creating and using a coordinate system allows you
to not only make quantitative measurements, but also provides an alternative
method of adding vectors.
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Choosing a coordinate system is similar to laying a grid drawn on a sheet of
transparent plastic on top of your problem. You have to choose where to put the
center of the grid (the origin) and establish the direction in which the axes point.
How do you choose the direction of the axes? There is never a single correct
answer, but some choices make the problem easier to solve than others. When the
motion you are describing is confined to the surface of Earth, it is often convenient
to have the x-axis point east and the y-axis point north. When the motion involves
an object moving through the air, the positive x-axis is often chosen to be horizontal
and the positive y-axis vertical (upward). If the motion if on a hill, it’s convenient to
place the positive x-axis in the direction of the motion and the y-axis perpendicular
to the x-axis.
After the coordinate system is chosen, the direction of any vector can be specified
relative to those coordinates. The direction of a vector is defined as the angle that
the vector makes with the x-axis, measured counterclockwise.
7. Resolve vectors into components using the sine and cosine functions.
A coordinate system allows you to expand your
description of a vector. In the coordinate system shown to
the right, the vector A is broken up or resolved into two
component vectors. One, Ax, is parallel to the x-axis, and
the other, Ay, is parallel to the y-axis. You can see that the
original vector is the sum of the two component vectors.
A
Ay
Ax
A = Ax + Ay
The process of breaking a vector into its components is sometimes called vector
resolution. The magnitude and sign of component vectors are called the
components. All algebraic calculations involve only the components of vectors, not
the vectors themselves. You can find the components by using trigonometry. The
components are calculated according to these equations, where the angle  is
measured counterclockwise from the positive axis.
Component Vectors
cos  =
adjacent side
, therefore Ax = A cos 
hypotenuse
sin  =
opposite side
, therefore Ay = A sin 
hypotenuse
When the angle that a vector makes with the x-axis is larger than 90° – that is, the
vector is in the second, third, or fourth quadrants – the sign of one or more
components is negative. Although the components are scalars, they can have both
positive and negative signs.
Sample Problem
A bus travels 23.0 km on a straight road that is 30.0° north of east. What are the
east and north components of its displacement?
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Sample Problem
A hiker walks 14.7 km at an angle 35.0° south of east. Find the east and north
components of this walk.
8. Apply the Pythagorean theorem and tangent function to calculate the magnitude
and direction of a resultant vector.
The Pythagorean Theorem states that for any right triangle, the square of the
hypotenuse – the side opposite the right angle – equals the sum of the squares of
the other two sides, or legs. The Pythagorean Theorem is applied to find the
magnitude of the resultant vector.
Pythagorean Theorem for Right Triangles
c2 = a2 + b2
(length of hypotenuse)2 = (length of one leg)2 + (length of other leg)2
In order to completely describe a resultant vector, you must also know the direction
of this vector. Because we have a right triangle, the tangent function can be used to
find the angle , which denotes the direction of the resultant vector.
Definition of the Tangent Function for Right Triangles
tan  =
opposite
adjacent
 opposite 

 adjacent 
  tan 1 
In the figure to the right, the two velocities of
the boat are represented by vectors. When
these two vectors are added, the resultant
velocity, vR, is 9.4 m/s at 32° north of east.
You can also think of the boat as traveling, in
each second, east 8.0 m and north 5.0 m at
the same time. Both statements have the
same meaning.
vR = 9.4 m/s, 32°
V2 = 5.0 m/s, 90°
A motorboat heads east at 8.0 m/s across a river that flows north at 5.0 m/s.
Starting from the west bank, the boat will travel 8.0 m east in one second. In the
same second it also travels 5.0 m north. The velocity north does not change the
velocity east. Neither does the velocity east change the velocity north. These two
perpendicular velocities are independent of each other. Perpendicular vector
quantities can be treated independently of one another.
V1 = 8.0 m/s, 0°
Suppose that the river is 80. meters wide. Because the boat’s velocity is 8.0 m/s
east, it will take the boat 10 seconds to cross the river. During this 10 seconds, the
boat will also be carried 50 meters downstream. In no way does the downstream
velocity change the velocity of the boat across the river.
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9. Add vectors that are not perpendicular.
Many objects move in one direction, and then turn at an acute angle before
continuing their motion. Because the original displacement vectors do not form a
right triangle, it is not possible to directly apply the tangent function or the
Pythagorean Theorem when adding the original two vectors.
Determining the magnitude and the direction of the resultant can be achieved by
resolving each of the displacement vectors into their x and y components. The
components along each axis can be added together. The magnitude of the
resultant can be found using the Pythagorean Theorem, and its direction can be
found using the tangent function.
Equilibrium - when two or more forces act concurrently on a body and the vector
sum (the resultant) is zero. An object that is at equilibrium is either stationary, or
moving at a constant velocity and not rotating.
Equilibrant - a single additional force that can be added to put a system of forces at
equilibrium if applied at the same point. The equilibrant is equal in magnitude but
opposite in direction to the resultant.
It is possible to begin with a single vector and think of it as the resultant of two
vectors. Usually, two new vectors are chosen that are perpendicular to each other.
These two vectors are called components of the original vector. Usually vectors are
broken down into either horizontal and vertical components or perpendicular and
parallel components. This is known resolving a vector into components.
The weight of an object on an inclined plane can be resolved into two perpendicular
components. One component F||, acts parallel to the plane; the other component,
F (normal force), acts perpendicular to the plane.
F||  FW  sin 


F
FW
F  FW  cos 
F||
Sample Problem
A 500 N crate is sitting on a 10o incline. What amount of force must be exerted to
keep this crate from sliding down the incline?
References: Holt Physics, pages 83-97, 106-119.
Tutorial on vectors: www.physicsclassroom.com/Class/vectors/
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