Converting a Vector Equation of a Plane to a Scalar Equation

[Chelly0704]

how does one convert a vector equation of a plane to a scalar equation. This is given the parametric equations of the plane

 

[radou]

Define the components of your vectors in the vector equation and apply the rules of vector multiplication.

 

[HallsofIvy]

Parametric equations of a plane are of the form x= as+ bt+ c, y= ds+ et+ f, z= gs+ ht+ k for number a, b, c, d, e, f, g, h, and k and parameters s and t.

The corresponding vector equation is $\vec{r}(t)= (as+ bt+ c)\vec{i}+ (ds+ et+ f)\vec{j}+ (gs+ ht+ k)\vec{k}$.

A vector in the "s" direction in that plane is $a\vec{i}+ d/vec{j}+ g\vec{k}$ and a vector in the "t" direction in that plane is $b\vec{i}+ e\vec{j}+ h\vec{k}$.

Their cross product, $(dh-eg)\vec{i}- (ah- bg)\vec{j}+ (ae-bd)\vec{k}$ is normal to the plane and (c, f, k) is a point in the plane so the scalar equation for the plane is (dh- eg)(x- c)- (ah- bg)(x- f)+ (ae- bd)(z- k)= 0.