Chapter 13.5, Problem 81E

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

  a. The line r = ⟨3, −1, 4⟩ + t ⟨6, −2, 8⟩ passes through the origin.

  b. Any two nonparallel lines in ℝ³ intersect.

  c. The plane x + y + z = 0 and the line x = t, y = t, z = t are parallel.

  d. The vector equations r = ⟨1, 2, 3⟩ + t ⟨1, 1, 1⟩ and R = ⟨1, 2, 3) + t ⟨−2, −2, −2⟩ describe the same line.

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

  f. Any two distinct lines in ℝ³ determine a unique plane.

  g. The vector ⟨−1, −5, 7⟩ is perpendicular to both the line x = 1 + 5t, y = 3 − t, z = 1 and the line x= 7t, y = 3, z = 3 + t.