Equation of a plane which parallel to a line

• Aug 19th 2007, 01:59 AM
moheelave4
Equation of a plane which parallel to a line

Find the equation of the plane through the line
r=(1,-3,4)+t(2,1,1)
and parallel to the line
r=s(1,2,3)

:confused::confused::confused:
• Aug 19th 2007, 04:01 AM
Rebesques
Call it (p).

The line r=(1,-3,4)+t(2,1,1) is parallel to the vector (2,1,1). Since (p) contains r, the vector (2,1,1) is parallel to (p) also.

Now (p) is also parallel to r=s(1,2,3), so it is parallel to (1,2,3).

------To find its parametric equation, we use the fact that planes are generated by two linearly independent parallel vectors:

r=a(2,1,1)+b(1,2,3), for a,b real.

-----To find its equation in (x,y,z), we need a vector perpendicular to (p). For a perpendicular vector, we find the cross product of the two vectors parallel to (p).

So, a vector normal to the surface is (2,1,1)x(1,2,3)=(1,-5,1).

A point on (p) can be found by assigning a random value for t in the equation of the line r=(1,-3,4)+t(2,1,1), say t=0: r=(1,-3,4).

If (x,y,z) is a point (p), then (x,y,z)-(1,-3,4) is a vector parallel to the plane, and must be normal to (1,-5,1). We now have the equation of the plane: (1,-5,1)$\displaystyle \cdot$(x-1, y+3,z-4)=0 or (x-1)-5(y+3)+(z-4)=0.
• Aug 19th 2007, 06:45 AM
earboth
Quote:

Originally Posted by moheelave4

Find the equation of the plane through the line
r=(1,-3,4)+t(2,1,1)
and parallel to the line
r=s(1,2,3)

...

Hello,

call the plane p. Since p contains the given line the point A(1, -3, 4) belongs to the plane too. Now you need two linear independent vectors to span(?) the plane. These vectors are known to you. If $\displaystyle \vec x$ is the position vector of/to every point of the plane the equation of the plane is:
$\displaystyle p: \vec x = \left(\begin{array}{c}1 \\-3 \\ 4 \end{array} \right) + r \cdot \left(\begin{array}{c} 2 \\1 \\ 1 \end{array} \right) + s \cdot \left(\begin{array}{c} 1 \\ 2\\ 3 \end{array} \right)~, ~r,s~\in~\mathbb{R}$
• Aug 19th 2007, 07:10 AM
Rebesques
Oops, for the parametric, do as Earboth did and don't forget to add the point (:o)
• Mar 24th 2008, 01:51 PM
Arivaz
Quote:

Originally Posted by earboth
Hello,

call the plane p. Since p contains the given line the point A(1, -3, 4) belongs to the plane too. Now you need two linear independent vectors to span(?) the plane. These vectors are known to you. If $\displaystyle \vec x$ is the position vector of/to every point of the plane the equation of the plane is:
$\displaystyle p: \vec x = \left(\begin{array}{c}1 \\-3 \\ 4 \end{array} \right) + r \cdot \left(\begin{array}{c} 2 \\1 \\ 1 \end{array} \right) + s \cdot \left(\begin{array}{c} 1 \\ 2\\ 3 \end{array} \right)~, ~r,s~\in~\mathbb{R}$

Is this correct? I think when the question says the plane passes through the line, the line intersects the plane at one point, but the whole line is not contained in the plane... help?