Given vectors v(1)=(2,1,2) and v(2)=(0,-1,1), decide whether the subspace is a line or a plane through the origin. If it is a plane, compute a vector N that is perpendicular to that plane?

Question

Given vectors v(1)=(2,1,2) and v(2)=(0,-1,1), decide whether the subspace is a line or a plane through the origin.  If it is a plane, compute a vector N that is perpendicular to that plane?

anonymous · Answer

@helder_edwin if you have time, could you help?

helder_edwin · Answer

u have decide wether the vectors are linearly independent or not.

anonymous · Answer

okay... I put them into a matrix and reduced it to see.  Does that tell me if it is a line or a plane?

helder_edwin · Answer

yes. when u put them as rows and reduced the matrix the resulting matrix will tell u wether the original vector were linearly independent or not.

helder_edwin · Answer

if have two vectors left in the row-reduced matrix then the vector are independent.

anonymous · Answer

for some reason I thought you had to put them in as column vectors, but I still got they were independent.  so if there are two vectors then it is a plane, if it is one vector they are dependent and a line?

helder_edwin · Answer

yes.

it really doesn't matter if u put them as columns or rows. the number of pivots will be the same.

anonymous · Answer

so then once I found out it was independent, then how do I calculate a perpendicular vector? Can I use either vector as my X, because I am trying to solve the equation for N
$$(X-X _{0})  N=0$$
correct?

helder_edwin · Answer

the easy way is using the cross product from several variable calculus.

helder_edwin · Answer

do u know this?

anonymous · Answer

i do not remember the cross product

helder_edwin · Answer

the vector u r looking for would be
$$ \large u=\begin{vmatrix}
                     \vec{i} & \vec{j} & \vec{k}\ 2 & 1 & 2\ 0 & -1 & 1
                  \end{vmatrix} $$

helder_edwin · Answer

$$ \large u=(3,-2,-2) $$

anonymous · Answer

jyes that is what i got too, but i can figure out why the book would divide by -2, and have the answer (-3/2, 1, 1)

helder_edwin · Answer

u can use the one u and i got or any other parallel vector
$$ \large (-1/2)u=(-3/2,1,1) $$

anonymous · Answer

so there is no particular reason why they would use that instead? or is there a different why the perpendicular vector could be compute that would get you that instead?

helder_edwin · Answer

yes. there's no particular reason to use any of them.

for instance, i was taught to avoid (when possible) vectors with fractions. but, again, it is a matter of taste.

anonymous · Answer

thank you again for your expertise, i appreciate all your help.

helder_edwin · Answer

u r welcome

see u soon