Show that the following equations represent the same plane?

Question

$L1 : r = u[-3, 2, 4] + v[-4, 7, 1], u, r \in \mathbb{R}$

$L2 : r = s[-1, 5, -3] + t[-1, -5, 7] , s, t \in \mathbb{R}$

(Hint: Express each direction vector in the first equation as a linear combination of the direction vectors in the second equation.)

Even the hint doesn't make sense to me

Answer 1

The first plane passes through the origin and has normal vector $n1 = [-3, 2, 4] \times [-4, 7, 1]$. The second plane also passes through the origin, and has normal vector $n2 = [-1, 5, -3] -\times [1, -5, 7]$.

So, the two planes are equal if and only if $n1$ and $n2$ are parallel. You can do the calculations to check this.

This doesn't use the hint, but I think it's an easy approach.

Answer 2

You wrote: "Even the hint doesn't make sense to me"

There are many possible ways of computations but let's look at what was given as the hint: "Express each direction vector in the first equation as a linear combination of the direction vectors in the second equation."

So the hint is basically suggesting to do this:

• Are there some $s$ and $t$ such that $[-3,2,4]=s[-1,5,-3]+t[-1,-5,7]$?
• Are there some $s'$ and $t'$ such that $[-4,7,1]=s'[-1,5,-3]+t'[-1,-5,7]$?

You should be able to find those numbers by solving a linear system. (I will just say that $s=\frac{17}{10}$, $t=\frac{13}{10}$ and $s'=\frac{27}{10}$, $t'=\frac{13}{10}$ are the solutions.)

The you should use some stuff you have already learned about vector subspaces to see how this helps to relate $L_1$ and $L_2$.