# Find the correct Plane.

• Jul 26th 2011, 06:52 AM
grgrsanjay
Find the correct Plane.
Find the equation of the plane passing through the points (2,2,-1) and (3,4,2) and parallel to a line whose direction ratios are (7,0,6)

• Jul 26th 2011, 07:04 AM
Siron
Re: Find the correct Plane!!
A plane is determined by three points or by two direction vectors and a point, now you've given one direction vector and two points, so you need to find another direction vector.

EDIT:
Do you want a vector equation or a cartesian equation?
• Jul 26th 2011, 11:43 AM
Siron
Re: Find the correct Plane!!
• Jul 26th 2011, 11:58 AM
Plato
Re: Find the correct Plane!!
Quote:

Originally Posted by grgrsanjay
Find the equation of the plane passing through the points (2,2,-1) and (3,4,2) and parallel to a line whose direction ratios are (7,0,6)

A plane is parallel to a line if the direction of the line is perpendicular to the normal of the plane.

$\displaystyle <1,2,3>\times <7,0,2>=~?$
• Jul 27th 2011, 05:09 AM
grgrsanjay
Re: Find the correct Plane!!
Quote:

Originally Posted by Siron

I need a Vector equation, I even need the steps please!
• Jul 27th 2011, 05:30 AM
Siron
Re: Find the correct Plane!!
This is the way I would do it:
Given a plane $\displaystyle \alpha$ and two points $\displaystyle P_1 (2,2,-1) ,P_2 (3,4,2) \in \alpha$ and the plane is parallel to a line who's direction ratios are $\displaystyle (7,0,6)$. In general the vector equation of a plane through one point $\displaystyle P_1$ and with two directions vectors $\displaystyle R,L$is: $\displaystyle P=P_1+kR+mL$. Now we already have (we work now in a three dimensional space):$\displaystyle (x,y,z)=(2,2,-1)+k(7,0,6)+mL$. Can you determine the other direction vector $\displaystyle L$?
.
• Jul 27th 2011, 06:24 AM
Plato
Re: Find the correct Plane!!
Quote:

Originally Posted by grgrsanjay
I need a Vector equation, I even need the steps please!

Define vectors:
\displaystyle \begin{align*} D&= <1,2,3> \\ P &=<2,2,-1>\\ R &=<x,y,z>\\ L&=<7,0,6>\end{align*}
Note $\displaystyle D=P_2-P_1$

Now the equation you want is
$\displaystyle (L\times D)\cdot (R-P)=0$
• Jul 28th 2011, 07:41 AM
grgrsanjay
Re: Find the correct Plane!!
Quote:

Originally Posted by Siron
This is the way I would do it:
Given a plane $\displaystyle \alpha$ and two points $\displaystyle P_1 (2,2,-1) ,P_2 (3,4,2) \in \alpha$ and the plane is parallel to a line who's direction ratios are $\displaystyle (7,0,6)$. In general the vector equation of a plane through one point $\displaystyle P_1$ and with two directions vectors $\displaystyle R,L$is: $\displaystyle P=P_1+kR+mL$. Now we already have (we work now in a three dimensional space):$\displaystyle (x,y,z)=(2,2,-1)+k(7,0,6)+mL$. Can you determine the other direction vector $\displaystyle L$?
.

I Couldn't Understand it completely....But the Good thing is that i understood fully what Plato Explained

So i think my problem is solved,
Thanks
• Jul 28th 2011, 09:32 AM
Siron
Re: Find the correct Plane!!
Ok :D, do you want to show me your solution?