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

Re: Vector Equation of a Plane

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**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 $\displaystyle u~\&~v$.

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 $\displaystyle (x_0, y_0, z_0)$.