Parametric form of a plane given a point and perpendicular vector

My apologies if a question like this already exists but I haven't been able to find anything. A point is given (1,4,5) and a perpendicular vector is also given (7,1,4). I am asked to find the parametric (vector) form of a plane using this information.

I understand that the parametric form is given by $$\ \vec r=\vec r_0+s \vec u +t \vec v; \ s,t \ \epsilon \ \Bbb R$$ and that in this case $$\vec r_0$$ will be (1,4,5). I was able to find the Cartesian equation to be $$7x+y+4z=31$$ and the answer for the parametric form is given as $$\ \vec r=(1,4,5)+s(4,0,-7)+t(0,4,-1)$$

I appreciate any help that anyone could give. Thanks

• You can have an infinite number of correct parametric form. Choose any two linearly independent vectors, both orthogonal to (7,1,4). A simple and quick way is to choose one coordinate of each of the vectors 0, as it is done in the answer. – user376343 Jun 9 '19 at 9:31
• @user376343 thanks, much appreciated – AgentWindu Jun 9 '19 at 9:48

In $$\ \vec r=\vec r_0+s \vec u +t \vec v$$, $$\vec r_0$$ is the position vector of a point on the plane, $$\vec u$$ and $$\vec v$$ are two vectors parallel to the plane. There are infinitely many choices for each one of them. $$\vec r_0=(1,4,5)$$ is a convenient choice. Both $$\vec u$$ and $$\vec v$$ should be perpendicular to $$(7,1,4)$$, and they should be linear independent. $$\vec u=(4,0,-7)$$ and $$\vec v=(0,4,-1)$$ are convenient choices as $$(7,1,4)\cdot(4,0,-7)=0$$ and $$(7,1,4)\cdot(0,4,-1)=0$$.
$$(1,-7,0)$$ is also an easy choice. Actually, we can take arbitrarily two coordinates and decide the third coordinate by the dot product. For example, if we take $$x=1$$ and $$z=-2$$, then $$(7,1,4)\cdot(x,y,z)=0$$ will yield $$y=1$$. $$(1,1,-2)$$ can also be a candidate for $$\vec u$$ or $$\vec v$$.
$$\vec u(x,y,z)$$ and $$\vec v(x',y',z')$$ are non-colinear vectors parallel to the plane, i.e. $$\vec n\cdot\vec u=\vec n\cdot\vec v=0$$, where $$\vec n=(7,1,4)$$ is a vector normal to the plane. We have$$\vec n\cdot\vec u=7x+y+4z=0$$and similarly for $$\vec v$$. You need to find suitable values of $$x,y,z$$ that satisfy the above.