PHYS 2108 Lecture Notes - Lecture 3: Net Force, Hypotenuse, Jeep Wrangler
3-1
LAB 3:
Vectors
Q: Whee is the estoo i Niholso Hall?
A: About 50 feet from the lab.
This ase is isuffiiet. It’s oe thig to ko that the e’s oo o the oe’s oo is a etai
distance from the lab, but that information is useless without a specified direction (especially in light of
the to oos eig o diffeet floos due to soe oloful ahitetue in the building).
Two categories exist for physical quantities.
Scalars are quantities that are just simply a number. They add and subtract like numbers and
they are probably quantities in which you are familiar. Energy, mass, time and temperature are
all examples of scalar quantities.
Vectors are quantities that have a direction. Velocity and acceleration are vectors, as are forces,
momentum and torque. As you progress further in physics we will find out that fields (electric,
magnetic and gravitational) are also vectors. Vectors do not simply add like numbers and
require very special geometric treatment.
Vectors contain two elements: a magnitude and a direction. The agitude is a sala ad tells us ho
uh stuff the vector is made from, and where the vector is pointed. The directional nature of a vector
is represented in the way we write its symbol: by giving it an arrow as in vector
.
We tend to wrongly interchange some words that are comparable in English, but are distinctly different
i siee like the diffeee etee peisio ad aua. Fo eaple, velocity is not the
same thing as speed. Speed is a number, like 60mph. Velocity is a speed with a direction, like 60mph
South. Speed itself is only the magnitude of the velocity. Additionally, displacement is not the same
thing as distance. Displacement is the net change of location from beginning to end. It has a magnitude
(in this case: a distance) and a direction. Pay attention to the words you use in describing physical
quantities. If the word is a vector, be sure to include its direction when you talk about it in reports and
problem solutions.
Components
A vector is made up of components, that is to sa piees that lie i opletel idepedet dietios.
For something two-dimensional, it has a component in the x-direction and y-direction. In the real world
we also consider a z-component. In a more exotic reality vectors can technically have components in
any number of multiple dimensions, but contemplating such foreign existences can hurt our fragile
minds.
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3-2
Consider a vector
that is 10 units long and oriented at 36.8699° above the x-axis (left figure). In
thinking about the components, make the vector the hypotenuse of an imaginary right triangle and put
the x- and y-components (vx and vy) on the sides of the triangle (middle figure). Finally, the length of the
y-component is the same whether we leave it on the edge of the triangle or slide it back to the origin
ight figue. The stadad epesetatio of a eto’s opoets is the laout i the ightost
figure, but until you are comfortable working with them you may find the middle figure most useful in
understanding the mathematics.
Copoets ae kid of a set of tu-by-tu dietios fo ho to eah the ed of a eto. The -
and y-components of this 10 unit-long vector are obvious because this is a special triangle with perfect
sides, and because we can count the squares. The components are 8 units to the right, and 6 up.
Typical etos ae’t so eas ad ust e oke ito opoets tigooetially. In understanding
trig functions, some definitions about the anatomy of a triangle are required. In the middle figure, the x-
component, vx, lies right next to the agle θ ad is alled the adjacent side. The component vy lies
across the tiagle fo the agle θ and is called the opposite side. Let’s t a fe tigooet
formulas and see what happens:
As ou gai oe epeiee, ou’ll e ale to eogize hih opoet is which and skip directly to
the equations for vx and vy just from looking at the sketch. Until then, draw out the triangle and start
ith a guess of si θ ad os θ ith opposites ad adjaets to see hih oespods to hih. The x
component will not always e a os θ, ad ie esa fo y. It will depend on which angle is labeled in
the tiagle if e had hose to easue θ fo the -axis instead of the x-axis, they would flip).
We can work in reverse order and find the magnitude and angle if we know the components. If the vx
and vy opoets ae legs i a tiagle, e a use Pthagoea’s Theoe to fid the hpoteuse ad
use some trig functions to discover the angle.
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3-3
Usig the figue o the peious page to idetif the loatio of θ, e a hoose opoets that ae
opposite and adjacent.
Use the definition of a tangent.
Take the inverse tangent to
isolate the agle θ.
Technically, you can use sines or cosines to find the angle also, but they each use the hypotenuse. If you
already know the components, it’s uh oe diet to sipl use the taget. I sua:
Eqn. (3-1)
Magnitude of a Vector
Eqn. (3-2)
Orientation of a Vector
(if measured from x-axis)
CAUTION: A calculator is a fairly unsophisticated device that can be confused by directions.
For example, if both the x- and y-components are negative, the calculator will
give you a small positive value for the angle. Always draw a picture of your two
components and the resultant vector in order to identify exactly where the
agle θ lies, and clearly describe it in the vector.
Representing a Vector
There are several ways to represent a vector in a way that expresses both its magnitude and direction.
For the 10-uit eto e’e ee okig ith, it a e epeseted in Magnitude and Direction
Notation as:
10 @ + 36.8699°
All angles measured starting
from the +x-axis and may be
anything from 0 - 360°
10 @ 36.8699° above the +x-axis
Specify the angle as measured
from any + or – axis.
10 @ 53.1301° to the right of the +y-axis
10 @ 36.8699° North of East
Treat the axes like a compass
rose and specify your
reference.
10 @ 53.1301° East of North
Another system involves representing vectors by their components, Unit Vector Notation.
o
Each component is explicitly
stated and the direction is
indicated by i-hat, j-hat or k-
hat. Some books use x,y,z
hats, othes use i,j,k hats.
A shorthand is to simply use
angled brackets.
o
< 8, 6 > or < 8, 6, 0 >
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Document Summary
Vectors (cid:862)whe(cid:396)e is the (cid:396)est(cid:396)oo(cid:373) i(cid:374) ni(cid:272)holso(cid:374) hall? (cid:863) Scalars are quantities that are just simply a number. They add and subtract like numbers and they are probably quantities in which you are familiar. Energy, mass, time and temperature are all examples of scalar quantities. Velocity and acceleration are vectors, as are forces, momentum and torque. As you progress further in physics we will find out that fields (electric, magnetic and gravitational) are also vectors. Vectors do not simply add like numbers and require very special geometric treatment. Vectors contain two elements: a magnitude and a direction. The (cid:373)ag(cid:374)itude is a s(cid:272)ala(cid:396) a(cid:374)d tells us (cid:862)ho(cid:449) (cid:373)u(cid:272)h stuff(cid:863) the vector is made from, and where the vector is pointed. The directional nature of a vector is represented in the way we write its symbol: by giving it an arrow as in vector .
