# Scalar equation of a plane

1. Apr 11, 2010

### Chelly0704

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

2. Apr 12, 2010

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

3. Apr 12, 2010

### 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.