Please help me decide an equation for a plane

Sep 2017
2
0
Sundsvall
Decide an equation in general form for a plane that is parallel to the vectors u and v and goes thru the point P=(1, 0, 3).

Vectors: u=(1, -2, 0) v=(3, -5, 1)

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If the terminology is incorrect it is because I have tried to translate it from Swedish:
"Ange, på normalform, en ekvation för det plan som är parallellt med vektorerna u och v och går genom punkten P=(1, 0, 3)"

I really dont get what I need to do. This is the last of six problems I need to solve for my homework that is due on Thursday. Please help me with this!

Thanks for reading!
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
A plane can be written as \(\displaystyle A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0\) where <A, B, C> is a vector perpendicular to the plane and \(\displaystyle (x_0, y_0, z_0)\) is a point on the plane. You are already given the point (1, 0, 3) so you only need to determine a vector perpendicular to the plane. Since the plane is perpendicular to the two given vectors, the cross product of the two vectors, which is perpendicular to the two vectors, is perpendicular to the plane. That is, take A, B, and C to be the components of the cross product of u and v and take \(\displaystyle x_0= 1\), \(\displaystyle y_0= 0\), \(\displaystyle z_0= 3\).
 

Plato

MHF Helper
Aug 2006
22,490
8,653
Decide an equation in general form for a plane that is parallel to the vectors u and v and goes thru the point $P=(1, 0, 3)$.

Vectors: $\vec{u}=<1, -2, 0>~\&~ \vec{v}=<3, -5, 1>$
OR
If $\vec{R}=<x,y,z>$ the the plane is $(\vec{u}\times\vec{v}){\cdot}(\vec{R}-<1,0,3>)=0$