Homework Sourse6 Give a parametrization for each of the following a the lin
6.) Give a parametrization for each of the following:
a.) the line passing through (1,2,3) and (-2,0,7)
b.) the graph of f(x)=x^2
c.) the square with vertices (0,0), (0,1), (1,1) and (1,0) (break it up into line segments)
d.) the ellipse given by (x^2/9) + (y^2/25) = 1
Solution
a) line passing through (1 , 2 , 3) and (-2 , 0 , 7)
let P = (1 , 2 , 3) and Q = (-2 , 0 , 7)
==> PQ = Q - P = <-2 , 0 , 7> - <1 , 2 , 3>
==> PQ = <-3 , -2 , 4>
Hence line through P and Q is r(t) = P + t(PQ)
==> r(t) = <1 , 2 , 3> + t<-3 , -2 , 4>
==> r(t) = <1-3t , 2 - 2t , 3 + 4t>
b) y = x2
parametrising x = t ==> y = t2
==> r(t) = <t , t2>
c) Vertices are (0 , 0) , (0 , 1) , (1 , 1) and (1 , 0)
let S1 be side with vertices (0 , 0) and (0 , 1) ==> S1 = <0 , 0 + t> = <0 , t> ; 0 <=t <= 1
S2 be side with vertices (0, 1) and (1 , 1) ==> S2 = <0 + t , 1> = <t , 1> ; 0 <= t <= 1
S3 be side with vertices (1 , 1) and (1 , 0) ==> S3 = <1 , t - 1> ; 0 <= t <= 1
S4 be side with vertices (0 , 0) and (1 , 0) ==> S4 = <0 +t , 0> = <t , 0> ; 0 <= t <= 1
d) x2/9 + y2/25 = 1
==> x2/32 + y2/52 = 1
general epplise equation is x2/a2 + y2/b2 = 1
==> x = acost , y = bsint
==> x = 3 cost , y = 5sint
Hence r(t) = < 3cost , 5 sint>
