Span{U,V} As A Plane: Linear Dependencies, Points, Cross Products, And Equations

sometimes Span{u,v} is a plane

linearly independent

If Span{u, v} is a plane, then it means that the two vectors u and v lie on the same plane. In other words, any linear combination of u and v can be expressed as a vector that lies on the same plane.

Here are some possible reasons why Span{u,v} is a plane:

1. Linearly dependent: If u and v are linearly dependent, then it means that one of them can be expressed as a scalar multiple of the other. For example, if u = 2v, then Span{u, v} is a line, since one vector is a scalar multiple of the other. However, if u and v are not linearly dependent, then Span{u, v} is a plane.

2. Two points on a plane: If u and v are two points on a plane, then any linear combination of u and v will also lie on the same plane. For example, if u = (1,0,0) and v = (0,1,0), then Span{u, v} is the x-y plane.

3. Cross product: If u and v are not parallel, then their cross product u x v will be a vector orthogonal to the plane containing u and v. We can then take the cross product of u and u x v or v and v x u to get another vector orthogonal to the plane. These three vectors (u, v, and one of the cross products) will form a basis for the plane.

4. Equation: If we have an equation of a plane in the form ax + by + cz = d, then any two linearly independent vectors u and v that satisfy the equation can be used to generate the plane. For example, if we have the equation x + y + z = 1, then u = (1,0,0) and v = (0,1,0) will both satisfy the equation and generate the plane.

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