# parametric equation of line.

• Oct 14th 2011, 07:58 PM
Kuma
parametric equation of line.
I have a normal vector (2,-2,0) of the a plane and a point on a line orthogonal to the plane, (4,0,1). So my goal is to try and find the parametric description of this line.
now to do that I need a vector v, but how can I find this vector with the given info?

the parametric representation is given by p + tv where p is a point and v is a vector. I need another point on this line to find the vector. Since this line is orthogonal to the plane I know that this vector should be parallel to the normal vector of the plane. But im having difficulty trying to find another point on this line.
• Oct 14th 2011, 08:50 PM
Deveno
Re: parametric equation of line.
maybe i'm missing something, here....why doesn't (4,0,1) + t(2,-2,0) work?
• Oct 15th 2011, 10:34 AM
Kuma
Re: parametric equation of line.
Well the line and the normal vector to the plane are parallel, but how do we know that the normal vector lies on the line?

I'm a bit confused myself. How can you just say (4,0,1) + t(2,-2,0) is the equation?
• Oct 15th 2011, 10:47 AM
Plato
Re: parametric equation of line.
Quote:

Originally Posted by Kuma
Well the line and the normal vector to the plane are parallel, but how do we know that the normal vector lies on the line?
I'm a bit confused myself. How can you just say (4,0,1) + t(2,-2,0) is the equation?

A line is perpendicular to a plane if and only if the line is parallel to the normal of the plane. In other words, the normal to the plane is the direction vector of the line.
Given a plane $\displaystyle \Pi: N\cdot(<x,y,z>-P)=0$ and a line $\displaystyle \ell:Q+tD$ then to say $\displaystyle \ell\bot\Pi$ means $\displaystyle D=\alpha N$ where $\displaystyle \alpha$ is a non-zero scalar.
• Oct 15th 2011, 11:00 AM
Deveno
Re: parametric equation of line.
the normal vector usually doesn't lie on the line going through a given point, it's just parallel to the line (there's a whole plane's worth of such lines, like a bundle of straws all lined up together).

but if we specify a point (any point) then we know which parallel line. "orthogonal to" and "normal" have almost the same meaning (usually, a normal vector is associated with some particular point in the plane). thus, every orthogonal vector to a plane (it's line direction) is parallel to a normal vector to a plane.