# How do I know if two planes are parallel?

• Oct 15th 2011, 07:36 AM
terrorsquid
How do I know if two planes are parallel?
Determine whether the given planes are parallel:

$4x-y+2z=5$
$7x-3y+4z=8$

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...

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

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

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

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

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

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

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

Can I solve this using an augmented matrix maybe?
• Oct 15th 2011, 07:45 AM
Plato
Re: How do I know if two planes are parallel?
Quote:

Originally Posted by terrorsquid
Determine whether the given planes are parallel:
$4x-y+2z=5$
$7x-3y+4z=8$

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

Are $<4,-4,2>~\&~<7,-3,4>$ parallel?
• Oct 15th 2011, 07:58 AM
terrorsquid
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?
• Oct 15th 2011, 08:08 AM
Plato
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.