Answer to Question #91381 in Analytic Geometry for Cassie

Question

Assume there are two planes Π1 and Π2 for which there exists a line of intersection L.
a) Suppose you find two different parametric forms for L using two different methods. How can you prove they are the same line?
b) Let there be vectors n1 and n2 that are normals to Π1 and Π2 respectively. If m = n1×n2 (cross product) and m is parallel to L, give a geometric description of this result.

Answer 1

a) If two different equations of the line are

$$(x_1, y_1, z_1)+t_1(a, b, c)\ \ \ (1)$$

and

$$(x_2, y_2, z_2)+t_2(d, e, f)\ \ \ (2)$$

where t₁ and t₂ are parameters.

Since there is only one line containing given point and parallel to given non-zero vector, it is enough to prove that both lines contains the same point and their directional vectors are parallel and therefore have proportional coordinates:

$$a/d=b/e=c/f$$

If equations (1) and (2) represent the same line, point (x_2, y₂, z₂) should satisfy equation (1)

$$(x_1, y_1, z_1)+t_1(a, b, c)=(x_2, y_2, z_2)$$

so system

$$x_1+at_1=x_2, y_1+bt_1=y_2, z_1+ct_1=z_2$$

should have unique solution t₁.

In general to prove that two equations represent the same curve it is enough to find a bijection between t₁ and t₂, so that corresponding point of curves are the same.

In case of ф straight line if a bijection exists, then it can be found from the equation

$$x_1+at_1=x_2+dt_2$$

or

$$t_2=(x_1-x_2)/d+at_1/d$$

and should satisfy equations

$$y_1+bt_1=y_2+et_2\ and\ z_1+ct_1=z_2+ft_2$$

b)The line of intersection L belongs to two planes Π1 and Π2, hence L is orthogonal to normal vectors n1 and n2.

Since planes Π1 and Π2 are not parallel, their normal vectors are not parallel either.

A direction of vector orthogonal to two given linearly independent vectors is uniquely determined and the vector m = n1×n2  is orthogonal to vectors n1 and n2, hence a directional vector of L is parallel to m.

Another geometric description is as follows:

Vector m = n1×n2 is orthogonal to vectors n1 and n2, hence m is parallel to Π1 and Π2, therefore m is parallel to the line L of intersection of planes.