Find plane which parallel to two vectors $L_{1} ( 3,1,10)$ and $L_{2}(1,-1,1)$ passes through a point $M(7,-10,3)$

Question

I`m trying to find a plane which parallel to two vectors $L_{1} ( 3,1,10)$ and $L_{2}(1,-1,1)$ passes through a point $M(7,-10,3)$
what I tried to do is to create $L_{1}L_{2}$ vector then to create the plane from $L_{1}L_{2}$ and the point $M$ but I think I did wrong.
Any suggestions?
Thanks!

Answer 1

You could first find a normal vector to the sought after plane.

Your sought after plane is parallel to the plane containing both of the vectors $L_1$ and $L_2$. So, a vector normal to your plane is any vector, $\bf n$, perpendicular to both $L_1$ and $L_2$.

There are at least two ways to find $\bf n$:

$\ \ \ 1)$ Find a nonzero solution ${\bf n}=(a,b,c)$ to the system of equations ${\bf n}\cdot L_1={ 0}\,; \ {\bf n}\cdot L_2={ 0}$.

or

$\ \ \ 2)$ Take the cross product of $L_1$ and $L_2$.

Once you've found $\bf n$, then you can use the formula giving the equation of the plane with normal vector $ (a,b,c)$ that passes through the point $(x_0,y_0,z_0)$: $$ a(x-x_0)+b(y-y_0)+c(z-z_0)=0. $$