vectors a vector quantity has both magnitude (size) and direction a scalar quantity only has size...
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Vectors
A vector quantity has both magnitude (size) and direction
A scalar quantity only has size (i.e. temperature, time, energy, etc.)
Parts of a vector:
headtail
length – represents the magnitude
We can perform math operations with vectors!
Vector AdditionA motor boat is moving 15 km/hr relative to the water.
The river current is 10 km/hr downstream.
How fast does the boat go (relative to the shore)
upstream and downstream?
Boat Upstream Vector
Vector AdditionA motor boat is moving 15 km/hr relative to the water.
The river current is 10 km/hr downstream.
How fast does the boat go (relative to the shore)
upstream and downstream?
Boat Upstream Vector
Boat Downstream Vector
Vector AdditionA motor boat is moving 15 km/hr relative to the water.
The river current is 10 km/hr downstream.
How fast does the boat go (relative to the shore)
upstream and downstream?
Current Vector = 10 km/hr downstream
Boat Upstream Vector
Boat Downstream Vector
Boat Velocity UpstreamUpstream: Place vectors head to tail,
net result, 5 km/hr upstream
bas
Boat Velocity UpstreamUpstream: Place vectors head to tail,
Boat VelocityUpstream: Place vectors head to tail,
net result, 5 km/hr upstream
Start
Finish
Difference
Boat VelocityDownstream: Place vectors head to tail,
Boat VelocityDownstream: Place vectors head to tail,
net result,
Boat VelocityDownstream: Place vectors head to tail,
net result, 25 km/hr downstream
abba
Commutative law
bas
Forces On An AirplaneWhen will it fly?
Gravity
Propulsion
Net Force?
Forces On An AirplaneWhen will it fly?
Gravity
Propulsion
Net Force
Plane Dives to the Ground
Forces On An AirplaneWhen will it fly?
Gravity
Propulsion
Lift
Net Force?
FrictionWhen will it fly?
Gravity
Propulsion
Lift
Net Force = 0 up or down
Plane rolls along the runway like a car because of propulsion.
Forces On An AirplaneWhen will it fly?
Gravity
Propulsion
Lift
Net Force
Plane Flies as long as Lift > Gravity
FrictionWhen will it fly?
Gravity
Propulsion
Lift
Air Resistance
Net Force = 0
Equilibrium
FlightWhen will it fly?
Gravity
Propulsion
Lift
Air Resistance
Net Force
Plane Flies as long as Lift > Gravity
AND Propulsion > Air Resistance
Vector Components
A component of a vector is the projection of the vector on an axis
x
y
A
6A
Magnitude, size is:
We can write the vector A as the sum of an x-component and y-component:
yAxAA yx
Ax , Ay = the x and y components of the vector A
x hat and and y hat are the unit vectors
xA
yA
x
y
A
adjAx
oppAy
yAxAA yx
If we only know the mag. ofA, and the angle, it makeswith the x-axis, how do we find the x, and y components?
sinsin
coscos
AAA
A
hyp
opp
AAA
A
hyp
adj
xy
xx
x
y
A
adjAx
oppAy
yAxAA yx
If we only know the x andy components, how can we find the magnitude of A?
22yx AAA
This comes from Pythagorean’s theorem
GO TO HITT
Adding (and subtracting) vectors by components
yBxBB
yAxAA
yx
yx
Let’s say I have two vectors:I want to calculate the vectorsum of these vectors:
yBAxBABA yyxx
yxyBxBB
yxyAxAA
yx
yx
85
43
Let’s say the vectors have the following values:
x
y
yxyBxBB
yxyAxAA
yx
yx
85
43
A
B
yx
yx
yBAxBABA yyxx
122
8453
x
y
A
BOur result is consistentwith the graphical method!
What’s the magnitudeof our new vector?
2121481444
122 22
.
BA
x
y
A B+
How would you find the angle, , the vector makes with the y-axis?
yxBA
122
y
x
opp = 2
adj = 12
01 5961
61
122
.tantan adj
opp
GO TO HITT
Multiplying vectors by scalars:
yAaxAaAa
yAxAA
yx
yx
So if the vector A was:the scalar, a = 5 then the new vector:
yxA
43 and it was multiplied by
yx
yxAa
2015
4535
Scalar Product: (aka dot product):
cosbaba
mag. of amag. of b
angle betweenthe vectors
Scalar Product: (aka dot product):
cosbaba
vectors scalars
The dot product is the productof two quantities:(1) mag. of one vector(2) scalar component of the second vector along the direction of the first
zzyyxx
zyxzyx
bababa
zbybxbzayaxaba
)()(
Go To HITT
Vector Product (aka cross product)The vector product produces a new vector who’s magnitudeis given by:
sinbabac
The direction of the new vector is given by the, “right hand rule”
Mathematically, we can find the direction usingmatrix operations.
zbybxbb
zayaxaa
zyx
zyx
zyx
zyx
bbb
aaa
zyx
ba
The cross productis determined from three determinants
zyx
zyx
bbb
aaa
zyx
ba
The determinants are used to find the components of the vector
1st : Strike out the first column and first row!
3rd : Strike out the 2nd column and first row
4th : Cross multiply the four components,subtract, andmultiply by -1:
2nd : Cross multiply the four components – and subtract:
yzzy baba x - component
zyx
zyx
bbb
aaa
zyx
ba
xzzx baba y - component
zyx
zyx
bbb
aaa
zyx
ba
5th: Cross out the last column and first row
6th : Cross multiply and subtract four elements
xyyx baba z-component
So then the new vector will be:
zbabaybabaxbababac xyyxxzzxyzzy
We’ll look more at the scalar product when we talk about angular momentum.
Example:
yxb
yxa
24
32
zx
zyx
zyx
zyx
ba
1616
1240124
432240024322
024
032
Example:
yxb
yxa
24
32
z
zyx
zyx
zyx
ba
16
12400
432240022003
024
032
Notice the resultant vector is in the z – direction!