Vectors and Direction
Investigation Key Question:
How do you give directions
in physics?
Vectors and Direction
• A scalar is a quantity
that can be completely
described by one
value: the magnitude.
• You can think of
magnitude as size or
amount, including
units.
Vectors and Direction
• A vector is a quantity
that includes both
magnitude and direction.
• Vectors require more
than one number.
– The information “1
kilometer, 40 degrees east
of north” is an example of a
vector.
Vectors and Direction
• In drawing a vector as
an arrow you must
choose a scale.
• If you walk five meters
east, your
displacement can be
represented by a 5 cm
arrow pointing to the
*use a ruler, not the boxes
east.
Vectors and Direction
• Suppose you walk 5 meters
east, turn, go 8 meters north,
then turn and go 3 meters west.
• Your position is now 8 meters
north and 2 meters east of
where you started.
• The diagonal vector that
connects the starting position
with the final position is called
the resultant.
Vectors and Direction
• The resultant is the sum of two
or more vectors added together.
• You could have walked a
shorter distance by going 2 m
east and 8 m north, and still
ended up in the same place.
•
• The resultant shows the most
direct line between the starting
position and the final position.
C
R
B
A
R = A+B+C
*Use a ruler not the boxes on graph paper!
Representing vectors with
components
• Every displacement vector in
two dimensions can be
represented by its two
perpendicular component
vectors.
• The process of describing a
vector in terms of two
perpendicular directions is
called resolution.
Representing vectors with
components
• Cartesian coordinates are also known as x-y
coordinates.
– The vector in the east-west direction is called the
x-component.
– The vector in the north-south direction is called the
y-component.
• The degrees on a compass are an example of
a polar coordinates system.
• Vectors in polar coordinates are usually
converted first to Cartesian coordinates.
Adding Vectors
• Writing vectors in components make it easy to add
them.
Subtracting Vectors
• To subtract one vector from another vector,
you subtract the components.
Calculating the resultant
vector by adding components
An ant walks 2 meters West, 3 meters
North, and 6 meters East. What is the
displacement of the ant?
1. You are asked for the resultant vector.
2. You are given 3 displacement vectors.
3. Sketch, then add the displacement vectors
by components.
4. Add the x and y coordinates for each
vector:
–
–
X1 = (-2, 0) m + X2 = (0, 3) m + X3 = (6, 0) m
= (-2 + 0 + 6, 0 + 3 + 0) m = (4, 3) m
–
The final displacement is 4 meters east and 3
meters north from where the ant started.
Calculating Vector
Components
• Finding components
graphically makes use
of a protractor.
• Draw a displacement
vector as an arrow of
appropriate length at
the specified angle.
• Mark the angle and use
a ruler to draw the
arrow.
Finding components
mathematically
• Finding components using trigonometry
is quicker and more accurate than the
graphical method.
• The triangle is a right triangle since the
sides are parallel to the x- and y-axes.
• The ratios of the sides of a right triangle
are determined by the angle and are
called sine and cosine.
Y
X
Finding the Magnitude of a
Vector
• When you know the x- and y- components of a
vector, and the vectors form a right triangle, you
can find the magnitude using the Pythagorean
theorem.
Adding Vectors
Algebraically
1. Make a chart
Vector
X
Y
2. Find the x- and ycomponents of all the
A = (r, Θ) = rcosΘ = rsinΘ
vectors
3. Add all of the numbers in
the X column
B = (r, Θ) = rcosΘ = rsinΘ
4. Add all of the numbers in
the Y column
5. This is your resultant in R = A + B A + B A + B
x
x
y
y
rectangular coordinates.
What Quadrant?
• Your answer for Θ is not necessarily
complete!
– If you have any negatives on your Rx or Ry,
you need to check your quadrant.
(+,+) = 1st = 0-90o
(-,+) = 2nd = 90o – 180o
(-,-) = 3rd = 180o -270o
(+,-) = 4th = 270o -360o
Equilibriant
• Like “equilibrium”
• The vector that is equal in magnitude,
but opposite in direction to the resultant.
• Ex.
R = (30m, -50o)
E = (30m, 130o)
Forces in Two
Dimensions
Investigation Key Question: How do
forces balance in two dimensions?
Force Vectors
• If an object is in
equilibrium, all of the
forces acting on it are
balanced and the net
force is zero.
• If the forces act in two
dimensions, then all of
the forces in the xdirection and y-direction
balance separately.
Equilibrium and Forces
• It is much more
difficult for a
gymnast to hold his
arms out at a 45degree angle.
• Why?
Forces in Two
Dimensions
1) Resolve the force supported by the
left arm into the x and y components.
2) Use the y-component to find the force in
the gymnast’s arms.
Forces in Two
Dimensions
The vertical force supported by the left arm must be
350 N because each arm supports half the weight.
(Fy = 350)
Resultant
• The force in the right arm must also be 495
newtons because it also has a vertical component
of 350 N.
Forces in Two
Dimensions
• When the gymnast’s arms
are at an angle, only part of
the force from each arm is
vertical. (350 N)
• The resultant force must be
larger (495 N) because the
vertical component in each
arm is only part of the
resultant.
The inclined plane
• An inclined plane is a straight surface,
usually with a slope.
Fn
• Consider a block
sliding down a ramp. Ff
• There are four forces
that act on the block:
–
–
–
–
gravity (weight).
Normal force
friction
the reaction force acting
on the block.
Fg
Fa
Forces on an inclined
plane
• The friction force
is equal to the
coefficient of
friction times the
normal force in the
y direction:
Ff = -mFn cosθ
Fn = mg
Ff = -mmg cosθ.
Motion on an inclined
plane
• Newton’s second law can be used to
calculate the acceleration once you
know the components of all the forces
on an incline.
• According to the second law:
Acceleration
(m/sec2)
a=F
m
Force (kg . m/sec2)
Mass (kg)
Motion on an inclined
plane
• Since the block can only accelerate along the
ramp, the force that matters is the net force in the
x direction, parallel to the ramp.
• If we ignore friction, and substitute Newton's' 2nd
Law and divide by m, the net force in the x
direction is:
Fx = m g sin θ
F
= ma
a = g sin θ
Motion on an inclined
plane
• To account for friction, the acceleration is reduced
by the opposing force of friction:
Fx = mg sin θ - m mg cos θ
Calculating acceleration
A skier with a mass of 50 kg is on a hill
making an angle of 20 degrees. The
friction force is 30 N. What is the skier’s
acceleration?
Fx = (50 kg)(9.8 m/s2) (sin 20o) = 167.6 N
Fnet = Fx – Ff = 167.6 N – 30 N = 137.6 N
Calculate the acceleration: a = F/m
a = 137.6 N ÷ 50 kg = 2.75 m/s2
Robot Navigation
• A Global Positioning System
(GPS) receiver determines
position to within a few
meters anywhere on Earth’s
surface.
• The receiver works by
comparing signals from three
different GPS satellites.
• About twenty-four satellites
orbit Earth and transmit radio
signals as part of this
positioning or navigation
system.