How do I know if two planes are parallel?

Determine whether the given planes are parallel:

$\displaystyle 4x-y+2z=5$

$\displaystyle 7x-3y+4z=8$

I think I can find a vector that is parallel to each plane, but I am not quite sure how to compare the vectors that I find. Here is what I have done:

I tried to find 2 points on each that satisfied the equations then found the vector joining the two points...

$\displaystyle 4(1)-(-1)+2(0) = 5 \rightarrow (1,-1,0)$

$\displaystyle 4(0)-(-1)+2(2) = 5 \rightarrow (0,-1,2)$

$\displaystyle \vec{v} = <1,0,-2>$

$\displaystyle 7(0)-3(0)+4(2) = 8 \rightarrow (0,0,2)$

$\displaystyle 7(1)-3(1)+4(1) = 8 \rightarrow (1,1,1)$

$\displaystyle \vec{r} = <-1,-1,1>$

How do I check to see if $\displaystyle \vec{v}$ and $\displaystyle \vec{r}$ are parallel?

Can I solve this using an augmented matrix maybe?

Re: How do I know if two planes are parallel?

Quote:

Originally Posted by

**terrorsquid** Determine whether the given planes are parallel:

$\displaystyle 4x-y+2z=5$

$\displaystyle 7x-3y+4z=8$

** Two planes are parallel if and only if their normals are parallel.**

Are $\displaystyle <4,-4,2>~\&~<7,-3,4>$ parallel?

Re: How do I know if two planes are parallel?

That's my question, I don't know how to tell if two vectors are parallel or not.

Surely the vectors that lie on each plane could be compared?

Re: How do I know if two planes are parallel?

Quote:

Originally Posted by

**terrorsquid** That's my question, I don't know how to tell if two vectors are parallel or not. Surely the vectors that lie on each plane could be compared?

You really should not try these questions until you master the fundamentals.

Two vectors are parallel if they are **non-zero scalar multiples of one another**.