Page 376 - 35Linear Algebra

376 Movie Scripts

Lets simplify this by calling V = (x, y, z) the vector of unknowns and N =
3
(a, b, c). Using the dot product in R we have
N V = d .

Remember that when vectors are perpendicular their dot products vanish. I.e.
U V = 0 ⇔ U ⊥ V . This means that if a vector V 0 solves our equation N V = d,
then so too does V 0 + C whenever C is perpendicular to N. This is because
N (V 0 + C) = N V 0 + N C = d + 0 = d .

But C is ANY vector perpendicular to N, so all the possibilities for C span
a plane whose normal vector is N. Hence we have shown that solutions to the
equation ax + by + cz = 0 are a plane with normal vector N = (a, b, c).

Pictures and Explanation

This video considers solutions sets for linear systems with three unknowns.
3
These are often called (x, y, z) and label points in R . Lets work case by case:
• If you have no equations at all, then any (x, y, z) is a solution, so the
3
solution set is all of R . The picture looks a little silly:

• For a single equation, the solution is a plane. This is explained in
this video or the accompanying script. The picture looks like this:

• For two equations, we must look at two planes. These usually intersect
along a line, so the solution set will also (usually) be a line:

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