# Vector equation of a plane

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• Mar 28th 2011, 03:35 PM
TN17
Vector equation of a plane
Write a vector and parametric equation for a plane that:
b) contains (-5,9,-3) and is parallel to [x,y,z] = [1, -2, 7] + s[4, -1, -3] and
[x,y,z] = [7,-2,15] + t[1,6,-8]

I'm not sure where to start. Usually, when asking for parallel lines, I find if the direction vectors are scalar multiples of each other, then I find out if s and t have the same value for all x y and z.

I'm confused about planes.
• Mar 29th 2011, 06:11 AM
TheEmptySet
Quote:

Originally Posted by TN17
Write a vector and parametric equation for a plane that:
b) contains (-5,9,-3) and is parallel to [x,y,z] = [1, -2, 7] + s[4, -1, -3] and
[x,y,z] = [7,-2,15] + t[1,6,-8]

I'm not sure where to start. Usually, when asking for parallel lines, I find if the direction vectors are scalar multiples of each other, then I find out if s and t have the same value for all x y and z.

I'm confused about planes.

One way to find the equation of a plane is to know a normal vector to the plane and a point in the plane. Since we have the point we need to find a normal vector to the plane. Since the plane must be parallel to both of the above lines we can find a vector perpendicular to both of them by taking the cross product of the direction vectors.

$\displaystyle \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & 7 \\ 4 & -1 & -3\end{vmatrix}=13\mathbf{i} +31\mathbf{j}+ 7\mathbf{k}$

Can you finish from here?