Find vector, parametric and scalar equations for the plane that contains the line r= (-1,3,1) + t(2,4,5) and also the point (2,0,-3)

Question

please include question and answer

Answer 1

Amanda:

One way to think about this problem is to find three points on the plane and then determine the equation of the plane containing these three points by cross product.

We know (2,0,-3) is a point.

Let t = 0 then (-1,3,1) is also a point.

Let t =1 then (-1+2,3+4,1+5) = (1,7,6) is also a point.

Let A= (2,0,-3) B=(-1,3,1) C=(1,7,6)

vector(AB) = < -1-2, 3-0,1+3> = <-3, 3, 4>

vector(AC) = <1-2,7-0,6+3> = <-1,7,9>

Taking the cross product <-3,3,4> X <-1,7,9> = <-1,23,-18>

So the equation of the plane will be -(x-2)+23*(y-0)-18*(z+3)=0

Or -x + 23y -18z = 52. This is the "standard" equation of the plane.

We can use vector(AB) = <-3,3,4> and point A=(2,0,-3) to determine parametric equations of the plane.

x-2 = -3*t

y-0 = 3*t

z+3 = 4*t

I'll let you figure out the rest.