Mathematics with vectors
• When dealing with more than one
dimension, can have the same equation
come up multiple times.
Example: Position equation in three
dimensions yields
x(t )  x0  v0, x t  12 a x t 2
y (t )  y0  v0, y t  12 a y t 2
z (t )  z0  v0, z t  12 a z t 2
• In a situation with multiple objects and
multiple physical quantities, the number of
equations increases dramatically!
– Need a shorthand to simplify the process.
– Vectors provide this simplification.
• Note that vectors provide a shorthand to
see physical relationships, but never allow
for numerical answers!
Definition of vectors
• Vectors are a set of numbers that describe the
same physical quantity
– Number of numbers equals the number of dimensions
Example: Blood pressure – use both systolic and
diastolic to determine health
• Vectors are physical objects with magnitude and
direction
Example: Herd migration – number of animals and
direction of movement determines migration.
Components
• Magnitude of vector
depicted by length
• Direction by angle
• Components are
projection onto the
various axes
– Length along axis is
magnitude in that direction
– One component for each
axis
Shorthand Notation
x  t   x0  v0t  12 at 2
is shorthand for
x  x   x0  v0, xt  axt
1
2
2
y (t )  y0  v0, yt  12 a yt 2
z (t )  z0  v0, zt  12 azt 2
where letter subscripts indicate
projection axis
Addition/Subtraction
• Add and subtract vectors by components
– Notice that if B has a component in an opposite
direction of A, then components subtract.
A  B  ( Ax  Bx ) xˆ  ( Ay  By ) yˆ
•Subtract vectors by reversing direction of
subtracted vector and adding
–Reverse direction by changing sign on all
components.
Vector Multiplication
• Three types of vector multiplication
– scalar x vector = vector
– vector x vector = scalar
– vector x vector = vector
• Scalar (number) x vector simply
“stretches” vector
– Scalar multiplies each component equally
Scalar (Dot) Product
• vector x vector = scalar
– Product of magnitude of one vector times projected
magnitude of other
A  B  A( B cos )
 Ax Bx  Ay By
Vector (Cross) Product
• vector x vector = vector
– Two (perpendicular) vectors define plane
– Direction of product is at right angles to this plane
– Magnitude is product of perpendicular vectors
A  B  AB sin 
Ballistic Motion
• Ballistic motion is the most basic motion in two
dimensions.
– Only acceleration is that of gravity
– Only acts in one direction
– Motion is in two dimensions
As a projectile thrown upward at a non-vertical angle moves in a
parabolic path, at what point along its path are the velocity and
acceleration vectors for the projectile parallel to each other?
1.
2.
3.
4.
at the point just before the projectile lands
at the highest point
at the launch point
nowhere
As a projectile thrown upward at a non-vertical angle moves in a
parabolic path, at what point along its path are the velocity and
acceleration vectors for the projectile parallel to each other?
1.
2.
3.
4.
at the point just before the projectile lands
at the highest point
at the launch point
nowhere
Solving ballistic motion
problems
• Treat each direction independently
Example:
A field biologist is collecting specimens. She
spots a rare monkey 10 m up in a tree 35 m
away.Moving carefully, she fires a tranquilizer
dart at the monkey. Unfortunately, at the moment
the trigger is pulled, the monkey lets go. If the
dart leaves the gun at 45 m/s, does the dart hit the
monkey?
1) Find initial angle
Using trigonometry, get
tan  0 
h
d
h
d
10 m
 tan 1
35 m
 15.95
 0  tan 1
2. Find time for dart to travel horizontally to tree
Apply position equation to horizontal motion
d  v0, x t  v0 cos 0t
d
t
v0 cos 0

35 m
 45 ms  cos15.95
 0.81 s
3. Find dart’s distance above ground when it reaches tree
Use distance equation for vertical motion
y  v0, y t  12 a y t 2  v0 sin  0t  12 gt 2
 (45
m
s
)sin 15.95  0.81 s  
1
2
9.81   0.81 s 
m
s2
 6.79 m
4. Find distance of monkey above ground
y  h  12 a y t 2

 (10 m)  12 9.81
 6.79 m
m
s2

 0.81 s 
2
2
As a projectile thrown upward at a non-vertical angle moves in a parabolic
path, at what point along its path are the velocity and acceleration vectors
for the projectile perpendicular to each other?
1. at the point midway between the launch point
and the highest point
2. at the launch point
3. nowhere
4. at the highest point