parameter representation

[Zman15]

I am in the end of chapter two of three chapters in my math book.
I have understood most of the chapter. But there is one thing i don't understand anything of and that is parameter representation.

Task 2.34

A boat follows the curve given by [x = -2 + 3h]
and [y = 9 - 4h)

Lengths are in meters and time in seconds.

A) Draw the curve for t between 0 and 3.
B) Determine the speed vector

Task 2.35
A) Draw the line where x = 1 + 2h and y = -1 + 3h
B) Find a parameter representation of line through the points A (0,5) and B (2,3). Use parameter p.
C) Calculate the intersection point between the lines.

task 2.36
Find by calculation where line m in example 25 intersects the x axis and where the line intersects the y axis.

example 25
Draw the curv M in [x = -1 + 2h] and [y = 5-3h]
B) Find a parameter representation of line through the points A (0, -1) and B (1,1). Use parameter p.
C) Calculate the intersection point between the two curves.

Thanks for the help.

 

[Ishuda]

I am in the end of chapter two of three chapters in my math book.
I have understood most of the chapter. But there is one thing i don't understand anything of and that is parameter representation.

Task 2.34

A boat follows the curve given by [x = -2 + 3h]
and [y = 9 - 4h)

Lengths are in meters and time in seconds.

A) Draw the curve for t between 0 and 3.
B) Determine the speed vector

Task 2.35
A) Draw the line where x = 1 + 2h and y = -1 + 3h
B) Find a parameter representation of line through the points A (0,5) and B (2,3). Use parameter p.
C) Calculate the intersection point between the lines.

task 2.36
Find by calculation where line m in example 25 intersects the x axis and where the line intersects the y axis.

example 25
Draw the curv M in [x = -1 + 2h] and [y = 5-3h]
B) Find a parameter representation of line through the points A (0, -1) and B (1,1). Use parameter p.
C) Calculate the intersection point between the two curves.

Thanks for the help.

In a lot of cases, you can turn a parametrization around to a familiar x-y if that can be understood easier by you. Lets take
something like Task 2.34.
A snail crawls across a flat court yard starting at (x,y) = (1,2) and traveling 3 meters per hour in the x direction and 4 meters per hour in the y direction. Well, our equations become
(1) x = 1 + 3 h
(2) y = 2 + 4 h
where h is in hours and x and y are in meters.
(A) Draw the curve for h between 0 and 2.
First, when h is 0, x is 1 and when h is 2, then x is 7. So h between 0 and 2 is the same as x between 1 and 7.
Next, turn the equations around and put x (and y) in terms of h so we have
(3) h = (x-1)/3
(4) h = (y-2)/4
so
(x-1)/3 = (y-2)/4
Multiplying through by 12 we have
4 (x-1) = 4x - 4 = 3(y-2) = 3 y - 6
or, rearranging,
(5) y = 2 (2 x + 1) / 3
which may be easier to understand.

However, note that x and y intercepts still mean the same thing, it is just that you get a value of, in this case, h and have to plug it into the other equation. For example the x intercept is when x is zero, so we have, from (5) that y=2/3. From (1) [or (3)] we have that h=-1/3 when x = 0 and then from (2) that y=2/3 as it should be.


B) Determine the speed vector
This can be done by noticing that starting from the point (x,y)=(1,2) to the point 1 hour later at (4, 6) forms a right triangle with sides 3 and 4 [the coefficients of h]. Using the Pythagorean Theorem, we have that the snail has traveled $\displaystyle 5 = \sqrt{3^2 + 4^2}$ meters. So the snail is traveling 5 meters/hour