# Thread: Show span is a plane through origin

1. ## Show span is a plane through origin

Let v1=[x1,y1,z1] and v2=[x2,y2,z2] assume that v1 and v2 are linearly independent, show that span{v1,v2} is a plane passing through the origin. What is the equation of that plane?

I know to show its a plane we need to show there are 2 degrees of freedom but from there on im pretty lost any help would be grealy appreciated.

2. Originally Posted by ChrisBickle
Let v1=[x1,y1,z1] and v2=[x2,y2,z2] assume that v1 and v2 are linearly independent, show that span{v1,v2} is a plane passing through the origin. What is the equation of that plane?

I know to show its a plane we need to show there are 2 degrees of freedom but from there on im pretty lost any help would be grealy appreciated.
We can use some Calculus common knowledge to show that $v_1$ and $v_2$ are a plane.

1 vector in 3 space forms a line
2 vectors in 3 space at most forms a plane and forms a line if they aren't lin. ind.
Since we are given that $v_1$ and v_2 are lin. ind., we know they form a plane.

Example:
Suppose $<1,2,3>$ is a normal vector to our plane then our equation would be:

$1(x-x_1)+2(y-y_2)+3(z-z_2)=0$

Where $(x_3,y_3,z_3)$ is a point in the plane.

You are giving a point is the origin (0,0,0). All you need is a normal vector now.