# Vector Equation of a Plane

• Jun 2nd 2013, 03:31 AM
Jin0730
Vector Equation of a Plane
Find a vector perpendicular to both i+2j-k and 3i-j+k. Hence find the Cartesian equation of the plane parallel to both i+2j-k and 3i-j+k which passes through the point (2,0,-3).

I managed to solve the first one but not the parallel part.

Explanation with the aid of diagram is much appreciated. Thanks in advance
• Jun 2nd 2013, 03:50 AM
Plato
Re: Vector Equation of a Plane
Quote:

Originally Posted by Jin0730
Find a vector perpendicular to both i+2j-k and 3i-j+k. Hence find the Cartesian equation of the plane parallel to both i+2j-k and 3i-j+k which passes through the point (2,0,-3).

I managed to solve the first one but not the parallel part.

Explanation with the aid of diagram is much appreciated. Thanks in advance

If $\displaystyle u~\&~v$ are non-parallel vectors and $\displaystyle P: (a,b,c)$ then $\displaystyle (u\times v)\cdot<x-a,y-b,z-c>=0$ is a plane which contains $\displaystyle P$ and is parallel to $\displaystyle u~\&~v$.
• Jun 2nd 2013, 04:55 AM
HallsofIvy
Re: Vector Equation of a Plane
Different way of saying the same thing: if <A, B, C> is a vector perpendicular to both vectors u and v, then $\displaystyle A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0$ is a plane parallel to both u and v and passing through $\displaystyle (x_0, y_0, z_0)$.