Vectors
A study of motion will involve the introduction of a variety of
quantities that are used to describe the physical world.
Examples of such quantities include distance, displacement,
speed, velocity, acceleration, force, mass, momentum, energy,
work, power, etc. All these quantities can by divided into two
categories - vectors and scalars.
A vector quantity is a quantity that is fully described by both
magnitude and direction. On the other hand, a scalar quantity is
a quantity that is fully described by its magnitude.
Recall:
vector  velocity = 30 mph north
scalar  speed = 30 mph
Vector quantities are often represented by scaled vector
diagrams. Vector diagrams depict a vector by use of an arrow
drawn to scale in a specific direction.
N
head
tail
W
E
Scale: 1 cm = 10 mph
S
velocity = 30 mph north
Vectors can be directed due East, due West, due South, and due
North. But some vectors are directed northeast (at a 45 degree
angle); and some vectors are even directed northeast, yet more
north than east. Thus, there is a clear need for some form of a
convention for identifying the direction of a vector that
is not due East, due West, due South, or due North.
N
W
E
S
The direction of a vector is often expressed as a
counterclockwise angle of rotation of the vector about its "tail"
from due East. Using this convention, a vector with a direction of
40 degrees is a vector that has been rotated degrees in a
counterclockwise direction relative to due east.
90O
40o
180O
0O
270O
Example:
Draw a 60 m/s vector rotated counterclockwise 120o.
Example:
Draw a 60 m/s vector rotated counterclockwise 120o.
90O
120o
180O
0O
Scale
10 m/s
270O
Adding vectors in 1-Dimension
If vectors in the same direction, add the magnitude of
the vectors.
The sum of the vectors is called the resultant.
If the vectors is in opposite direction, subtract the
magnitude of the vectors.
Adding vectors in 2-Dimensions require a bit more work.
There are two methods to adding vectors in 2-Dimensions.
Method 1: Head-to-tail method
a
b
Copy the vectors and placed them in a head to tail
arrangement.
Adding vectors in 2-Dimensions require a bit more work.
There are two methods to adding vectors in 2-Dimensions.
Method 1: Head-to-tail method
b
a+b
a
Move vector a to the tail of vector b. The sum of the two
vectors is the vector connecting from tail of vector a to the head
of vector b.
Adding vectors in 2-Dimensions require a bit more work.
There are two methods to adding vectors in 2-Dimensions.
Method 2: Parallelogram method
a
b
Copy the vectors and connect them tail-to-tail.
Adding vectors in 2-Dimensions require a bit more work.
There are two methods to adding vectors in 2-Dimensions.
Method 1: Parallelogram method
a+b
b
a
Create a parallelogram by copying another set of vectors
(dotted). Connect the corner of the tails to the corner
with the two heads to create the resultant.
If the vectors are perpendicular to each other, then the sum of
the two vectors can be computed by using the Pythagorean
theorem.
To add non-perpendicular vectors, it would require trigonometry
and the use of sine and cosine to determine the horizontal and
vertical component of a vector.
Vector resolution
Any diagonal vectors (has an angle with the horizontal)
can be resolved into its horizontal and vertical components.
Draw a horizontal and vertical line to form a right
triangle.
Vector resolution
Any diagonal vectors (has an angle with the horizontal)
can be resolved into its horizontal and vertical components.
Make a vertical vector the same length as the height of the right triangle
and place it tail-to-tail with the original vector.
Make a horizontal vector along the base of the right triangle. The blue
vector and the red vectors are the components of the green vector.
Example:
Two forces are applied to slide a crate across the floor. A 30 N
force is applied in a northerly direction and a 50 N force is
applied in an easterly direction.
1. Draw a top view free-body diagram to scale of the forces on
the crate.
2. Draw a resultant (net force) on the free-body diagram using
either the head-to-tail method or the parallelogram method.
3. Calculate the net force on the crate using the Pythagorean
Theorem.
30 N
50 N
Scale
10 N
Head-to-tail method
30 N
resultant
50 N
Scale
10 N
Parallelogram method
30 N
50 N
Scale
10 N
Pythagorean Theorem
a2  b2  c2
c  a2  b2
c  30 2  50 2
c
a = 30 N
b = 50 N
Scale
10 N
c  900  2500
c  3400
c  58.31N