Finding a vector normal to the plane with position point and parallel to two vectors

Question

A cross product of the two parallel vectors will get the vector normal to the plane. But I'm looking for a specific normal vector.

The plane with point $A(3,2,1)$ and parallel to $u = 5i+2k$ and $v= 3i+j+2k$

The normal vector will need to be of the form $n = (a,4,b)$ so we will need to find the value of $a$ and $b$.

Answer 1

Exactly what StrangerLoop recommends you to do. As any scalar multiple of the normal vector will still be normal, you can multiply it with any constant (that is called scaling) and still have a normal vector. You wrote the cross product was (4,10,-1) ,without checking if that is correct you can now say that also the vectors (4/10,1,-1/10) and therefore (1.6,4,-0.4) are orthogonal to the plane

Answer 2

Use the cross product to find a vector $v$ normal to the plane. Any scalar multiple of this vector is also normal to the plane. So solve $c v = n$ for the scalar $c$, and you're done.

Answer 3

Since $\vec{n}$ will have to be orthogonal to both $\vec{u}$ and $\vec{v}$, you'll need $5a+2b=0$ and $3a+4+2b=0$. Those two equations will have a unique solution, allowing you to find $a$ and $b$.

An alternative would be to take $\vec{u}\times\vec{v}$ and scale it so that the resulting middle component is $4$.