Name,
Vectors & Parametric Equations
Pre-Cal H
_
Date.
_
Per
'-../L. Intro
Parametric equations are a good way to represent real-life motion that has both horizontal and
vertical components. We are only working in two dimensions in these models.
II.
Lines
To write a parametric
equation of a line using vectors, use the formula:
New position
= Initial
position (at t
= 0)
+
t· Direction (or velocity) vector
= (xo, Yo)+ t( 6X, 6Y)
(x,y)
Note: there are an infinite number of parametric
the same rectangular equation.
Example 1: A line passes through point A( -5,2)
equations for any line but they all simplify to
and point B(1,S). Write an equation of the line
through the points A and B.
Solution:
First find-th'e direction vector AB, which is (6,3). -Fhen (x,y)=(-S,2)+t(6,3).
x(t)=-S+6t
In parametric
~When
equations: { ( )
y t = 2+3t
you convert to rectangular
form (by removing the parameter),
you get y
=
_
III. The effect of gravity (objects that are thrown)
If an object is thrown with an initial velocity Vo (measured in feet per second) at an angle
(measured from the horizontal) and it started
by the following parametric
e
at an initial height Yo' then it can be represented
equations:
x(t)=(vocose)t
{ y(t) = -16t2 +(vo sine)t+ Yo
The term -16t2
is due to the effect
of gravity. g
= -32 f+/sec"
or g
= -9.8 meters
per second"
Example 2: A baseball player hits a ball with an initial velocity of 92 feet per second at an angle
of 55° from the horizontal. The initial height of the ball was 3 feet. Find the maximum height
attained and the total horizontal distance traveled by the ball. Will it clear a 10-foot fence that
is 240 feet from home plate?
x(t)
{ y(t)
= (92cosSS)t
= -16t
2
+ (92sin 55)t + 3
'Sketch a complete graph of the problem situation. Indicate
~For
the fence use 2nd-Draw #2 - Line (240, 0, 240, 10)
the window used. Mode: Par, Degr'ee
Example 2 (continued) What would happen if there was a 10 mph wind blowing in the opposite
direction of the horizontal path of the ball? Convert 10 mph =
feet per second.
IV. Circular Motion
A ferris wheel has a radius of 20 feet and makes one revolution (counterclockwise) every 12
seconds. find parametric equations to model the motion of the wheel. Assume the center of the
ferris wheel is located at the point (0, 20). Gravity is not affecting the motion of the wheel.
x(t)=
{ y(t) =
Youare -standi-n-gon-the ground at point D, a distance of 75 feet from the bottom of the ferris
wheel and your arm's initial height is at the same level as the bottom of the wheel. At the instant
when your friend is at point A, you throw a ball to her at 60 feet per second at an angle of 60°
above the horizontal. How close does the ball get to your friend?
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75 feet
D