Question

Explain why there are many different vector and parametric equations for a plane.
Provide specific examples to justify your answer. (3 marks)

True or False: Two lines must be parallel if they have the same
direction vectors.

Answer 1

A plane can be defined by two direction vector and a point on the plane.
The two direction vectors would "stabilize" the plane and you then move it parallel
to itself to go through the given point.
e.g.
Suppose you have a plane x + y + z = 10
3 points would be (1,0,9), (3,3,1) and (-2,5,7)
so two direction vectors would be <2, 3,-8> and <-5,2,6>
using the point (3,3,1) one possible vector equation would be
r = (3,3,1) + t(2,3,-8) + s(-5,2,6)
the corresponding parametrics are:
x = 3 + 2t - 5s
y = 3 + 3t 2s
z = 1 - 8t + 6s

of course we could have used one of the other two points, giving us two
more versions of equations for the plane
another set of 3 points could be (0,0,10), (-4, 8, 6) and (5,5,0)
we could find totally different direction vectors and using any of the points
we could find several different vector and parametric equations of the same plane

b) T or F:
What does the phrase "same direction vectors" mean to you

Answer 2

sorry, I have a typo
the point (3,3,1) is not on the plane I gave, should have been (3,3,4)
<b>Unfortunately</b>, this messes up both of the direction vector that I used.
<b>Fortunately</b> , my reasoning and argument are still valid, I am sure you can
correct my typo and the results that follow.

Answer 3

Thank you very much