The problem is:
In the equation of the plane: Ax+By+Cz+D=0 spanned in R^3 by a=(1,3,0) and b=(0,1,2) find the parameters A,B,C and D.
I don`t have any clue about how to solve this problem. Does anybody have a good suggestion?
My second problem is:
Let "a", "b" and "c" be vectors in R^3. If "a" belongs to the plane spanned by "b" and "c", can these three vectors be linearly independent?
My answer to this would be, no they are always linearly dependent because "a" can be written in any case as a combination of "b" and "c" - is my answer correct?
1. If you have a new question please start a new thread.
2. Your question belongs to Pre-Calculus.
3. Here are the steps you should do:
Determine . Use
Use the value of the limit to determine a: Solve for a.
Keep in mind that this composed function is not differentiable at x = 0.
I still don't know how to answer the first question:
In the equation of the plane: Ax+By+Cz+D=0 spanned in R^3 by a=(1,3,0) and b=(0,1,2) find the parameters A,B,C and D.
Maybe there exists a theorem or something which I don't know. The only thing I can see is that here the three vectors are dependent.
The vectors are dependent so there has to be a way to write a third vector as a linear combination of the other two.
x_{1}=1 x_{2}=0 x_{3}=?
y_{1}=3 y_{2}=1 y_{3}=?
z_{1}=0 z_{2}=2 z_{3}=?
But I don't know how to continue!
Since the problem says "spanned" by, the plane must be a subspace of and so must contain (0, 0, 0), the "0 vector".
The first thing we can do is divide through by, say, A and write the equation as x+ B'y+ C'z+ D'= 0. Setting x= y= z= 0 gives D'= 0. Set x=
1, y= 2, z= 0 and then x= 0, y= 1, z= 2 to get two equations for B' and C'.
Ah, now I understand a little bit better, thanks.
The vectors are not dependend because the R3-space contains at least three independent vectors, is this true?
But I don`t really understand how you got the values for A B C and D? Could you explain it a little bit more in detail please?
sorry today I am slow on the uptake!
What is the reasoning behind this calculation?
As HallsofIvy suggested we have to add the (0,0,0).
So we have a plan spanned by three vectors:
(1,3,0)
(0,1,2)
(0,0,0)
now depending on this three vectors we have to find values for the parameters in this equation:
Ax+By+Cz+D=0
Why is the cross product of the vectors the result of the equation?