Transcript unit vectors - thephysicsman

Vectors
• A vector is basically an arrow that
represents the magnitude and direction of a
measurement.
• The length of the vector represents its
magnitude.
Vector A
Vector B
If vectors A and B represent compatible quantities, i.e., displacement,
then the length of the vectors reveals that A is approximately twice as large as B.
• The direction of the arrow represents the
direction in whatever coordinate system is
in use.
A
Vector A is parallel to vector B, meaning they share the same direction.
B
A
A
Still parallel.
B
B
Anti-parallel:
meaning 180 degree
change in direction.
B
Vector A is perpendicular to vector B, meaning
they are at 90 degrees difference in direction.
A
B
Vector A is oblique to vector B, meaning
they are neither parallel, nor perpendicular
in relation to each other.
A
Notation
• A vector quantity is usually represented as
the variable in question with an arrow over
it. Sometime it may be written in boldface
rather than with an arrow.
𝐴
• The scalar value or magnitude that relates to
a vector (i.e., the length) can be represented
as the variable of the vector quantity
without the arrow (also not in boldface) or
as that variable placed inside an absolute
value bracket.
𝐴 or just A  easy to confuse
Coordinate systems
(How we write vectors and describe them)
Cartesian Coordinates
X and Y directions are rectilinear
Cartesian space in three dimensions
Cartesian space in two dimensions
Axes are interchangeable,
and can reference a y-z or
x-z plane as needed.
We will do most of our work in this course in 2D as once these
principles are mastered, adding the third dimension is more of the
same as far as the mathematics are concerned.
Remember that when referencing the Cartesian system
you say:
“Vertical axis” vs. “Horizontal axis.”
For a standard x-y plane, your function would be:
“y vs. x,”
or
“y with respect to x.”
Cartesian Coordinates
• We can represent a vector in Cartesian
coordinates by giving its end point (the tip
of the arrow) as an ordered pair (2D) or
ordered triple (3D)
(x,y)
(3,4)
(x,y,z)
(3,4,5)
Sketch These!
This notation is O.K., but less meaningful in a physics context
(my opinion). The idea of representing a vector this way is,
however, VERY useful in computer science and graphic animation.
Now, another method…
• We can represent a vector in Cartesian coordinates
by showing the Resultant as the sum of x,y, and z
component unit vectors.
• A unit vector is simply a vector of magnitude
(length) 1 unit in a given direction. In this case,
the x-, y-, and z-directions, respectively.
• Sometimes instead of x,y,z we instead use i,j,k to
represent the same thing.
This looks like:
Let’s use a vector representing a force acting
on an object, say the force of the wind on a
kite. If the wind was blowing with a force of
10 Newton’s (unit of force) North and 6 N to
the East we might write any of these.
6 𝑁 𝐸𝑎𝑠𝑡 + 10 𝑁 𝑁𝑜𝑟𝑡ℎ
6 𝑁 𝐸 + 10 𝑁 𝑁
6 𝑁 𝑥 + 10 𝑁 𝑦
Best
𝑅 = 6 𝑁 𝑥 + 10 𝑁 𝑦
R
6N
10 N
Notice how the vector is expressed as the
sum of force values multiplied by a unit
vector (literally a length of 1-unit in the
given direction)?
This is because the vector itself is, in fact,
the sum of these components, an element
we will use to our advantage in a later
lesson.
Polar Coordinates
Relate a radius (the magnitude of the vector)
and an angle of incline
Angles are always designated from the positive x-axis (or East, if you use
the cardinal points) unless otherwise indicated. This convention is
routinely employed in math and science, but not in navigation.
(r,)
(12 m, 30o)
12 m @ 30o
12 m @ 30o N of E
12 m @ 60o E of N
-12 m @ 30o S of W
-12 m @ 210o
Draw these vectors.
12 m
300
These are all the same vector!
Practice
Sketch the following vectors.
1) R = -3x + y
2) R = 10x
3) R = 3y
4) R = 2y + 6z
5) R = i + 3j
6) R = 2j – 4i
7) R = 4i – 2j
8) R = x + 2y + 3z
Practice (cont’d)
Sketch the following vectors
1) 30 m/s @ 15o N of E
7) 22 m/s @ -20o
2) 12 m E
9) 22 m/s @ 20o
3) 9.8 m/s2 down
10) r = 6m @ 35o
4) 50 m/s @ 130o
11) r = 3m @ 270o
5) 50 m/s @ 50o N of W
12) r = -9m
6) -13 m @ 10o E of S
Tools:
Pythagorean Theorm
a b  c
2
2
c  a b
2
2
2
Resolution of independent
vector components
As we have seen with our Cartesian notation, a vector
can be represented as the sum of its parts.
If we let the hypotenuse of a right triangle represent the
a vector, the legs of that triangle represent the horizontal
and vertical components of that vector.
This allows us to break a vector down to find out its
magnitude in the horizontal (x) and vertical (y)
directions.
Why do you think this would be important?
Why might it be important in physics to disassemble a
vector into its horizontal and vertical components?
Answer:
Because in physics we can utilize the principles of linear
motion to then analyze each part, or component of the
motion independently of the other.
• We use this technique in physics because, as you
will learn shortly, vectors of the same variable that
are at right angles to each other do not have any
effect on each other.
• That means that motion in the horizontal direction
does not have any effect on motion in the vertical
direction!
• While we make that statement, it remains
important to remember that the path of an object
(its displacement) depends on both components.
Unfortunately...
We need trigonometry to do this...
Side a corresponds to angle A
Side b corresponds to angle B
Side c corresponds to angle C
sin  = opp/hyp
cos  = adj/hyp
tan  = opp/adj
sin A = a/c
cos A = b/c
tan A = a/b
sin B = b/c
cos B = a/c
tan B = b/a
So if you know any two angles, and any two sides
you can extrapolate the rest of the triangle.
Example
If a car travels at 50 m/s at 20o North of East, find the
horizontal(east) and vertical (north) components
of the velocity.
Rx = Rcoso
Rx = 50cos20o
Ry= Rsino
Ry = 50sin20o
BE CAREFUL!!!
You will come to notice that we often
use sine for your “y” component and
cosine for your “x” component, but it
always depends on the orientation of the
given angle within the system.
Write out your trig defn’s and carefully
examine your figures EVERY TIME.
We won’t use this one much in class, but it is incredibly useful
in general geometry, particularly on SAT and ACT math. As a
bonus it is easy to remember.
The Law of Sines
sin A sin B sin C


a
b
c
This relationship will allow you to solve ANY triangle long as
you know at least 1 side and 2 angles, or 2 sides and 1 angle.
This is very handy for right triangles since you always know at
least 1 angle (90o) and have the Pythagorean theorem
available.
Adding vectors
(a preview of next lesson)
When adding vectors, place the 1st vector at the
origin.
Next place the tail end of one vector
to the head end of the other.
Then draw the resultant vector from the origin to
the tip of the second vector.
R
Hey that looks like the representation
of a Cartesian vector!

6N
10 N
More practice:
Relative velocities
1) Find the resultant velocity of a boat that crosses a river due
east @ 4 m/s while the current runs south @ 1 m/s.
2) What is the displacement of a plane that flies south for 3.0
hours at 500 km/h with a 20 km/h tailwind? A 15 km/h headwind?
3) A cannonball is shot upwards at an angle of 30o above the
horizontal with a velocity of 35 m/s. Find the horizontal and
vertical components of the velocity. Draw these component vectors.
4) A car drives down a street at 30 m/s. A man is walking in the
same direction as the car at 2 m/s as he passes a stationary mailbox.
What is the velocity of the car with respect to the man? The car
with respect to the mailbox? The man with respect to the car?
4) An evil physics student fires a potato gun
forward out of a truck traveling at 25 m/s.
If the gun propels the potato at 52 m/s,
how fast is the potato traveling
a) When it strikes a stationary parked car?
b) When it strikes a cyclist riding forward (in the
same direction) at 4 m/s?
c) When it strikes a cyclist riding “backwards”
relative to the truck at 4 m/s?
d) Now the potato was fired backwards off of
the truck. Do a), b) & c) for this case.
5) Train A heads east at 175 m/s. Train B
heads west at 150 m/s. What is the velocity
of train A with respect to train B?
6) The trains in problem 5 are now both
traveling north.
What is the velocity of train B with respect
to train A?
What is the velocity of train A with respect
to train B?