Question

) The two vectors u<-1,2,3> and v = <-2,3,2> determine a plane in space. Mark each of the vectors below as "T" if the vector lies in the same plane as u and "F" if not.

Answer 1

1. <3,0,0>
2. <6,-9,-6>
3. <-4,7,8>
4. <-3,4,1>

Answer 2

There are a couple of ways I can think of to solve this, what do you think might be some useful properties to use?

Answer 3

First i dont get what does that mean by saying " two vecotrs u<-1,2,3> and v=<-2,3,2> determine a plane in space
Does that mean they are on the planes or just two position vectors that determine the two points on the plane

Answer 4

Oh. It's saying that any two vectors that are linearly independent (not parallel) will define a plane.

Answer 5

Or rather, that those two vectors are not parallel and therefore define a plane

Answer 6

You can also picture the tips of those vectors as lying in the plane

Answer 7

so they are not parallel but "in the plane"

Answer 8

right. They are parallel to the plane, but not parallel to each other.

Answer 9

ok,,
i have thought of using cross product

Answer 10

Good idea. What will that tell you?

Answer 11

the normal vector of the plane

Answer 12

indeed. What does it mean to be normal to a plane?

Answer 13

any vectors parallel to the plane, but not necessarily in the plane, will have a 0 dot product with the the normal vector?

Answer 14

any vector parallel to the plane will be in the plane

Answer 15

and yes, all the vectors in the plane are orthogonal to the normal vector of the plane (the dot product of the vector and the normal will be 0)

Answer 16

why any vector parallel to the plane will be in the plane?

Answer 17

sorry this is a new idea to me...i am kinda dumb right now

Answer 18

Ok, since the plane is given by two vectors

Answer 19

it intersects the origin

Answer 20

because vectors start at the origin, and end on the point specified with <x,y,z>

Answer 21

all vectors are so?

Answer 22

yes, vectors are simply directions, not positions

Answer 23

So all the vectors in the plane will be orthogonal to the normal of the plane.

Answer 24

ok got it :) thanks~~~~!!!

Answer 25

oh i have one more question

Answer 26

ok

Answer 27

the textbook i am using talks about displacement vector, which is defined by the difference of two points in space, and it does not go through the origin... so are there different kinds of vectors?

Answer 28

No, vectors can be moved around

Answer 29

they have a magnitude and a direction, but not a position

Answer 30

But there are not really different kinds of vectors (except when you're talking about vectors in different vector spaces)

Answer 31

oh so you can always visualize a vector starting at the origin and points toward some direction,,,kk i think i get it now Thanks :)

Answer 32

yep =)