Answer:
x(t) = -1-2t
y(t) = 5-5t
z(t) = 3+4t
Explanation:
If you know the vector equation we can easily convert it to the parametric equation because we just need to rewrite the expression in terms of x,t,z of t:
r(t) = (−1−2t)i + (5−5t)j + (3+4t)k
x(t) = -1-2t
y(t) = 5-5t
z(t) = 3+4t
The important thing behind this concept is that you can represent a line in the space with a system of equations or with a vector notation and depending on what do you need to do you can change easily from one notation to the other. Note that the vector is made of a position vector (constant) and a direction vector in the direction of the line represented by the system of equations.
The rewritten form of the vector equation of r(t) is;
x = ¹/₅(2y - 15) = ¹/₂(1 - z)
We are given the vector equation;
r(t) = (−1 − 2t)i + (5 − 5t)j + (3 + 4t)k
The vector equation above is the parametric form of the line;
x(t) = (−1 − 2t)
y(t) = (5 − 5t)
z(t) = (3 + 4t)
Another way to represent the line is symmetric. Thus;
x = -1 - 2t
t = -¹/₂(x + 1)
Thus;
y = [5 − 5(-¹/₂(x + 1))]
y = 5 + ⁵/₂x + ⁵/₂
y = ¹/₂(15 + 5x)
x = ¹/₅(2y - 15)
similarly;
z = 3 + 4(-¹/₂(x + 1))
z = 3 - 2x - 2
z = 1 - 2x
x = ¹/₂(1 - z)
Combining the expressions of x gives us;
x = ¹/₅(2y - 15) = ¹/₂(1 - z)
Read more at; https://brainly.com/question/15403277
