Question

In a usually right-angled (x, y, z)-coordinate system, three planes are given by the following system of linear equations:


`\alpha_1: x-y+(a-1)z=a\\\alpha_2: x+(a-1)y=0 \\\alpha_3: ax+ay+(a-1)z=0`
a ∈ R

a) Solve the equation system for
`a=2`

b) Find the value of
`a` for which the three planes intersection is a straight line and specify a parametric equation for the line.

c) Find the value of
`a` for which the three plans have no common point.

d) Is there a value of
`a` for which the three plans have the common intersection
`(1, -(1)/(69), (4624)/(4761))`?

Please, explain how you found the solutions :-)

Extra:
e) Illustrate questions a) and b) with Maple software.
Hint: The
`implicitplot3d` command can sure to be helpful. Also note that the argument
`orientation` can lock your plot in a favorable position for inspection.

Answer 1

a) (x, y, z) = (1, -1, 0)

b) a = 0; L = {t, t, 0}

c) a = 1 . . or . . a = 3

d) No

Explanation:

a) The solution using row-reduction techniques is shown in the first attachment, part (a). The generic solution is shown evaluated for a=2, the result being (1, -1, 0).

You will note that the solution gives indeterminate values for a=1 and a=3, which make the denominators zero.

b) For this, we use the point solution of part (a) together with the cross product of the normal vectors of planes 1 and 2. Said cross product will be in the direction of their line of intersection. We substitute the parametric equation for that line into the equation for plane 3 and solve for a. The result is a = 0.

The simplification of the resulting equation also gives a=1 and a=3, but we treat those as extraneous solutions because the corresponding common point of intersection of the planes does not exist.

c) The values a = 1 and a = 3 were found in part (a). These are values of a that make the common point indeterminate.

d) Substituting the given values for (x, y, z) gives three equations for a that have 3 different solutions. Hence there cannot be a value of a that makes this be a common point of intersection.

e) The second attachment illustrates part (a), with a=2. The third attachment illustrates part (b) with a=0.

The fourth attachment illustrates part (c). The first picture has the triangular tunnel between the planes in a vertical orientation (a=1). The second picture has the triangular tunnel extending from lower front to upper back (a=3).