anonymous
  • anonymous
) 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.
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
1. <3,0,0> 2. <6,-9,-6> 3. <-4,7,8> 4. <-3,4,1>
anonymous
  • anonymous
There are a couple of ways I can think of to solve this, what do you think might be some useful properties to use?
anonymous
  • anonymous
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

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anonymous
  • anonymous
Oh. It's saying that any two vectors that are linearly independent (not parallel) will define a plane.
anonymous
  • anonymous
Or rather, that those two vectors are not parallel and therefore define a plane
anonymous
  • anonymous
You can also picture the tips of those vectors as lying in the plane
anonymous
  • anonymous
so they are not parallel but "in the plane"
anonymous
  • anonymous
right. They are parallel to the plane, but not parallel to each other.
anonymous
  • anonymous
ok,, i have thought of using cross product
anonymous
  • anonymous
Good idea. What will that tell you?
anonymous
  • anonymous
the normal vector of the plane
anonymous
  • anonymous
indeed. What does it mean to be normal to a plane?
anonymous
  • anonymous
any vectors parallel to the plane, but not necessarily in the plane, will have a 0 dot product with the the normal vector?
anonymous
  • anonymous
any vector parallel to the plane will be in the plane
anonymous
  • anonymous
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)
anonymous
  • anonymous
why any vector parallel to the plane will be in the plane?
anonymous
  • anonymous
sorry this is a new idea to me...i am kinda dumb right now
anonymous
  • anonymous
Ok, since the plane is given by two vectors
anonymous
  • anonymous
it intersects the origin
anonymous
  • anonymous
because vectors start at the origin, and end on the point specified with
anonymous
  • anonymous
all vectors are so?
anonymous
  • anonymous
yes, vectors are simply directions, not positions
anonymous
  • anonymous
So all the vectors in the plane will be orthogonal to the normal of the plane.
anonymous
  • anonymous
ok got it :) thanks~~~~!!!
anonymous
  • anonymous
oh i have one more question
anonymous
  • anonymous
ok
anonymous
  • anonymous
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?
anonymous
  • anonymous
No, vectors can be moved around
anonymous
  • anonymous
they have a magnitude and a direction, but not a position
anonymous
  • anonymous
But there are not really different kinds of vectors (except when you're talking about vectors in different vector spaces)
anonymous
  • anonymous
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 :)
anonymous
  • anonymous
yep =)

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