Write the parametric equations for the lines and planes?

Question

a). x = 3y - 7
b). x - 2y + 3z = 6

Answer 1

Write the parametric equations for the lines and planes.

a) x = 3y - 7
3y = x + 7
y = (1/3)x + 7/3

Slope = ∆y/∆x = 1/3
Directional vector v = <∆x, ∆y> = <3, 1> = 3i + j

Pick a point on the line. Let's choose P(-7, 0).
The equation of the line is:
L = P + tv
L = <-7, 0> + t<3, 1> = <-7 + 3t, t>
where t is a constant ranging over the real numbers

Parametrically we have:

x = -7 + 3t
y = t

b) x - 2y + 3z = 6

To write the equation of a plane in parametric form we need two directional vectors, u and v. The easiest way to get them is to pick three non-colinear points, P, Q, R.

P(6, 0, 0); Q(0, -3, 0); R(0, 0, 2)

u = PQ = = <-6, -3, 0>

Any non-zero multiple of u is also a directional vector. Divide by -3.

u = <2, 1, 0>

v = PR = = <0, 3, 2>

The equation of the plane can be written with a point and the two directional vectors u and v. Let's choose R(0, 0, 2).

Plane = R + su + tv
Plane = <0, 0, 2> + s<2, 1, 0> + t<0, 3, 2>
Plane = <2s, s + 3t, 2 + 2t>
where s and t are constants ranging over the real numbers

Parametrically we have:

x = 2s
y = s + 3t
z = 2 + 2t

Answer 2

(3/7)y-x/7 =1
x/6-y/3+z/2=1