At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
1. <3,0,0> 2. <6,-9,-6> 3. <-4,7,8> 4. <-3,4,1>
There are a couple of ways I can think of to solve this, what do you think might be some useful properties to use?
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
Oh. It's saying that any two vectors that are linearly independent (not parallel) will define a plane.
Or rather, that those two vectors are not parallel and therefore define a plane
You can also picture the tips of those vectors as lying in the plane
so they are not parallel but "in the plane"
right. They are parallel to the plane, but not parallel to each other.
ok,, i have thought of using cross product
Good idea. What will that tell you?
the normal vector of the plane
indeed. What does it mean to be normal to a plane?
any vectors parallel to the plane, but not necessarily in the plane, will have a 0 dot product with the the normal vector?
any vector parallel to the plane will be in the plane
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)
why any vector parallel to the plane will be in the plane?
sorry this is a new idea to me...i am kinda dumb right now
Ok, since the plane is given by two vectors
it intersects the origin
because vectors start at the origin, and end on the point specified with
all vectors are so?
yes, vectors are simply directions, not positions
So all the vectors in the plane will be orthogonal to the normal of the plane.
ok got it :) thanks~~~~!!!
oh i have one more question
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?
No, vectors can be moved around
they have a magnitude and a direction, but not a position
But there are not really different kinds of vectors (except when you're talking about vectors in different vector spaces)
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 :)