what does it mean for a vector to be in a plane? 4th edition, linear algebra and its application, gilbert strang
exercise 2.1 question 17


*Let P be the plane in R^3 with equation x+y-2z = 4. The origin (0,0,0) is not in P! Find two vectors in P and check that their sum is not in P
Answer at the back --> (4,0,0) is on the plane, (0,4,0) is on the plane but their sum (4,4,0) is not on the plane.


Questions: What does it mean for a vector to be in a plane? Does the end point of the vector being on the plane mean that the vector is in the plane?
This is a central theme in second chapter that strang has used wherein he called the necessity of passing through origin for a subspace as a consequence of closure under addition and closure under scalar multiplication. to quote something - "the distinction between a subset and a subspace is made clear by example. In each case can you add vectors and multiply by scalars, without leaving the space?" and he goes on to demonstrate with a few examples
(Can you ever add two vectors and leave the space, or perform scalar multiplication to the same result?)

Take the $x,y$ and $z$ coordinates of the vector, insert them for $x,y$ and $z$ in the equation for the plane, and see what you end up with. In this case we get $4+0-2\cdot0=4$. Is that equality correct? If so, the vector is on the plane. If not, then the vector is not in the plane.
Do the same thing for $(0,4,0)$ and $(4,4,0)$ and check whether those two are on the plane too.
