Question

Find in parametric form the equation for all lines passing through (1,1,1) and lying in the plane x + 2y - z = 2.

I read the answer key and still don't get how to do this problem.

Answer 1

x +2(y-1) -z =0

so ( 1, 1, 0 ) is a point on the plane

Answer 2

and ( 1,2,-1) is a normal to the plane

Answer 3

I am not 100% sure, I dont like linear algebra and all that , but I think if you cross those two vectors you will get a direction vector along of the edges of the plane

Answer 4

not sure to get the other direction vector though

Answer 5

The answer key points out that 1,1,1 is already on the plane. Thus, the line from any point on the plane through 1,1,1 must also be parallel to the plane. What confuses me is the answer:

x = 1 + at; y = 1 + bt; z = 1 + ct

I don't understand where the number one comes from in each of those answers.

Answer 6

yeh.. its fairly standard form of a line

X= position vector + (parameter ) ( direction vector )

Answer 7

(1,1,1) is a point on the plane, it can be used as a position vecotr for the line

Answer 8

and a,b, c are compnents of a direction vecotr , but there are actually two direction vecotrs to a plane , because you have the one position vector and then two non parallel vectors that you take the span of to produce the parallelogram

Answer 9

Thank you.