How would you write a vector equation for a line using parametric equations? For example:
A line is defined by x=-2-t and y=4+2t,teR. a) Write a vector equation for this line using parametric equations.
b) write a second vector equation different from the first
Also, R(-9,18) lies on the line
The vector equation for the line is
r(t) = x(t) i + y(t) j
= (-2-t)i + (4+2t)j
r(0) = -2i+4j
r(t) = -2i+4j + t(-i+2j)
To write a vector equation for a line using parametric equations, you can organize the given parametric equations into a single equation with vector notation. Here's how you can do it:
a) Given parametric equations:
x = -2 - t
y = 4 + 2t
To create a vector equation, we can combine the x and y equations into a single equation:
r(t) = <x, y> = <-2 - t, 4 + 2t>
So, the vector equation for the line using the given parametric equations is:
r(t) = <-2 - t, 4 + 2t>
b) To write a second vector equation for the line, we can use different parameterization. For example, we can use s instead of t:
r(s) = <-2 - s, 4 + 2s>
Note that the vector equation r(s) represents the same line as in part a, but with a different parameterization. This means that for any value of s, the coordinates in the vector equation will correspond to the same points on the line.