Why linearly independent vectors define a plane
In the above applet, the two vectors u and v, and the point (0,0,3) can be used to define the plane 3x + 2y + 4z = 12. Any point on this plane, can therefore be reached with a linear combination of the vectors u and v. You can manipulate the two sliders to change the values of a and b, and therefore move point A, which is defined as au + bv.