Plane in 3 dimensions (Please check my work)

**janvdl** · January 3rd 2009, 08:14 AM

Find scalar equations of the form $ax + by + cz = 0$ for the following plane in $R^3$ :

The plane through the points:

$P(0;1;1)$
$Q(1;0;1)$
$R(1;1;0)$

${PQ}^{\rightarrow} = (1;-1;0)$

${QR}^{\rightarrow} = (0;1;-1)$

The perpendicular vector $= {PQ}^{\rightarrow} \times {QR}^{\rightarrow}$

Then I just picked point $Q$ to substitute into the equation:

$1(x -1) + 1(y - 0) + 1(z - 1) = 0$

Multiplying it out:

$x + y + z = 2$

And that should be the equation of the plane?

**Mush** · January 3rd 2009, 08:25 AM

Originally Posted by janvdl

Find scalar equations of the form $ax + by + cz = 0$ for the following plane in $R^3$ :

The plane through the points:

$P(0;1;1)$
$Q(1;0;1)$
$R(1;1;0)$

${PQ}^{\rightarrow} = (1;-1;0)$

${QR}^{\rightarrow} = (0;1;-1)$

The perpendicular vector $= {PQ}^{\rightarrow} \times {QR}^{\rightarrow}$

Then I just picked point $Q$ to substitute into the equation:

$1(x -1) + 1(y - 0) + 1(z - 1) = 0$

Multiplying it out:

$x + y + z = 2$

And that should be the equation of the plane?

Correct.

**Opalg** · January 3rd 2009, 08:26 AM

That's fine. Notice that (1) x+y+z = 2 is a linear equation, so it represents a plane; (2) the equation is obviously satisfied when (x,y,z) is P, Q or R. So that has to be the right answer!

**HallsofIvy** · January 3rd 2009, 03:59 PM

As Opalq said, it is easy to check it yourself.
Do P: (0, 1, 1), Q: (1, 0, 1), and R: (1, 1, 0) all satisfy x+ y+ z= 2? If yes, that is the equation of the plane containing those three points and if not, it is not.

(In fact, you could have found the equation just by noting that the three components of each of those points sums to 2.)

**mr fantastic** · January 3rd 2009, 06:50 PM

Originally Posted by HallsofIvy

As Opalq said, it is easy to check it yourself.
Do P

0, 1, 1), Q

1, 0, 1), and R

1, 1, 0) all satisfy x+ y+ z= 2? If yes, that is the equation of the plane containing those three points and if not, it is not.

(In fact, you could have found the equation just by noting that the three components of each of those points sums to 2.)

or : ( ...... The space makes a difference if you don't disable the smilies ....

*ahem* or did I mean : )

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