Chapter 12.1, Problem 71E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    The plane passing through the point (1, 1, 1) with a normal vector n = 〈1, 2, –3〉 is the same as the plane passing through the point (3, 0, 1) with a normal vector n = 〈–2, –4, 6〉.

b.    The equations x + y – z = 1 and – x – y + z = 1 describe the same plane.

c.    Given a plane Q, there is exactly one plane orthogonal to Q.

d.    Given a line l and a point P₀ not on l, there is exactly one plane that contains l and passes through P₀.

e.    Given a plane R and a point P₀, there is exactly one plane that is orthogonal to R and passes through P₀.

f.    Any two distinct lines in ¡³ determine a unique plane.

g.    If plane Q is orthogonal to plane R and plane R is orthogonal to plane S, then plane Q is orthogonal to plane S.