Question

find the equation of the plane containing the line OR=(2,-1,4)^t+u(1,3,5)^t which is parallel to the line OR=(-1,4,0)^t+s(4,-2,1)^t?

Answer 1

your question is strangely phrased, what is this ^t stuff?

Answer 2

transpose i don't think that matters in this case

Answer 3

ok, and why do both lines have the same name? is that a mistake?

Answer 4

i was wondering the same thing, must be a mistake by the teacher.

Answer 5

are u and s just parameters?

Answer 6

yes

Answer 7

my hunch is to say that if the first line is in the plane, and the second line is parallel to the plane, then the cross-product of the two vectors (1,3,5) and (4,-2,1) should be the normal vector of the plane
if we know the normal vector of the plane, and a point on the plane (2,-1,4) we can get the equation of the plane

does this argument sound reasonable to you?

Answer 8

it sounds right because the two lines don't have the same direction so i don't see how they are parallel to one an other

Answer 9

it says that the *plane* containing the first line is parallel to the second, not that the two lines are parallel to each other
the wording is kinda screwy in my opion, plus the agreed upon typo

Answer 10

opinion*

Answer 11

that makes more sense now. thanks!! and yes these teachers love to trick us lol

Answer 12

so I say let's cross the two vectors (1,3,5) and (4,-2,1) to find the normal vector \(\vec n\)

Answer 13

just a question, why are we crossing those (1,3,5) with (4,2,-1)

Answer 14

because if both those vectors are parallel to the plane, then crossing them will produce a vector that is perpendicular to both, and therefor perpendicular to the plane
i.e. our normal vector, which we need to find the equation of the plane

Answer 15

I'm gonna call the vectors a and b, and the point on the plane P to avoid the confusing notation your teacher gave