7 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-17.

Answer Problems 1–30 without referring back to the text. Fill in the blank or answer true/false.

  1. The vectors 〈−4, −6, 10〉 and 〈−10, −15, 25〉 are parallel.
  2. In 3-space, any three distinct points determine a plane.
  3. The line x = 1 + 5t, y = 1 − 2t, z = 4 + t and the plane 2x + 3y − 4z = 1 are perpendicular.
  4. Nonzero vectors a and b are parallel if a × b = 0.
  5. If the angle between a and b is obtuse, a · b < 0.
  6. If a is a unit vector, then a · a = 1.
  7. The cross product of two vectors is not commutative.
  8. The terminal point of the vector a − b is at the terminal point of a.
  9. (a × b) · c = a · (b × c).
  10. If a, b, c, and d are nonzero coplanar vectors, then (a × b) × (c × d) = 0.
  11. The sum of 3i + 4j + 5k and 6i − 2j − 3k is .
  12. If a · b = 0, the nonzero vectors a and b are .
  13. (−k) × (5j) =
  14. i · (i × j) =
  15. −12i + 4j + 6k =
  16. =
  17. A vector that is normal to the plane −6x + y − 7z + 10 = 0 is .
  18. The plane x + 3y − z = 5 contains the point (1, −2, ).
  19. The point of intersection of the line x − 1 = (y + 2)/3 = (z + 1)/2 and the plane x + 2y − z = 13 is .
  20. A unit vector that has the opposite direction of a = 4i + 3j − 5k is .
  21. If = 〈3, 5, −4〉 and P1 has coordinates (2, 1, 7), then the coordinates of P2 are .
  22. The midpoint of the line segment between P1(4, 3, 10) and P2(6, −2, −5) has coordinates .
  23. If = 7.2, = 10, and the angle between a and b is 135°, then a · b = .
  24. If a = 〈3, 1, 0〉, b = 〈−1, 2, 1〉, and c = 〈0, −2, 2〉, then a · (2b + 4c) = .
  25. The x-, y-, and z-intercepts of the plane 2x − 3y + 4z = 24 are, respectively, .
  26. The angle θ between the vectors a = i + j and b = i − k is .
  27. The area of a triangle with two sides given by a = 〈1, 3, −1〉 and b = 〈2, −1, 2〉 is .
  28. An equation of the plane containing (3, 6, −2) and with normal vector n = 3i + k is .
  29. The distance from the plane y = −5 to the point (4, −3, 1) is .
  30. The vectors 〈1, 3, c〉 and 〈−2, −6, 5〉 are parallel for c = and orthogonal for c = .
  31. Find a unit vector that is perpendicular to both a = i + j and b = i − 2j + k.
  32. Find the direction cosines and direction angles of the vector a = i + j − k.

In Problems 33–36, let a = 〈1, 2, −2〉 and b = 〈4, 3, 0〉. Find the indicated number or vector.

  1. compba
  2. projab
  3. proja(a + b)
  4. projb(a − b)
  5. Let r be the position vector of a variable point P(x, y, z) in space and let a be a constant vector. Determine the surface described by (a) (r − a) · r = 0 and (b) (r − a) · a = 0.
  6. Use the dot product to determine whether the points (4, 2, −2), (2, 4, −3), and (6, 7, −5) are vertices of a right triangle.
  7. Find symmetric equations for the line through the point (7, 3, −5) that is parallel to (x − 3)/4 = (y + 4)/(−2) = (z − 9)/6.
  8. Find parametric equations for the line through the point (5, −9, 3) that is perpendicular to the plane 8x + 3y − 4z = 13.
  9. Show that the lines x = 1 − 2t, y = 3t, z = 1 + t and x = 1 + 2s, y = −4 + s, z = −1 + s intersect orthogonally.
  10. Find an equation of the plane containing the points (0, 0, 0), (2, 3, 1), (1, 0, 2).
  11. Find an equation of the plane containing the lines x = t, y = 4t, z = −2t, and x = 1 + t, y = 1 + 4t, z = 3 − 2t.
  12. Find an equation of the plane containing (1, 7, −1) that is perpendicular to the line of intersection of −x + y − 8z = 4 and 3x − y + 2z = 0.
  13. A constant force of 10 N in the direction of a = i + j moves a block on a frictionless surface from P1(4, 1, 0) to P2(7, 4, 0). Suppose distance is measured in meters. Find the work done.
  14. In Problem 45, find the work done in moving the block between the same points if another constant force of 50 N in the direction of b = i acts simultaneously with the original force.
  15. Water rushing from a fire hose exerts a horizontal force F1 of magnitude 200 lb. See FIGURE 7.R.1. What is the magnitude of the force F3 that a firefighter must exert to hold the hose at an angle of 45° from the horizontal?
    3 forces form a triangle. Vector F subscript 1 equals 200 i is horizontal. Vector F subscript 2 forms an angle of 45 degrees with F subscript 1. F subscript 3 joins the endpoints of F subscript 1 to F subscript 2.

    FIGURE 7.R.1 Fire hose in Problem 47

  16. A uniform ball of weight 50 lb is supported by two frictionless planes as shown in FIGURE 7.R.2. Let the force exerted by the supporting plane 1 on the ball be F1 and the force exerted by the plane 2 be F2. Since the ball is held in equilibrium, we must have w + F1 + F2 = 0, where w = −50j. Find the magnitudes of the forces F1 and F2. [Hint: Assume the forces F1 and F2 are normal to the planes 1 and 2, respectively, and act along lines through the center C of the ball. Place the origin of a two-dimensional coordinate system at C.]
    A plane P subscript 1 goes up and to the left at 45 degrees from horizontal. Another plane P subscript 2 goes up and to the right at 30 degrees from horizontal. A circle with center C is tangent to both planes. A downward arrow from C is labeled w. Normal vectors to P subscript 1 and P subscript 2 point towards C.

    FIGURE 7.R.2 Supported ball in Problem 48

  17. Determine whether the set of vectors 〈a1, 0, a3〉 under addition and scalar multiplication defined by

    is a vector space.
  18. Determine whether the vectors 〈1, 1, 2〉, 〈0, 2, 3〉, and 〈0, 1, −1〉 are linearly independent in R3.
  19. Determine whether the set of polynomials in Pn satisfying the condition d2p/dx2 = 0 is a subspace of Pn. If it is, find a basis for the subspace.
  20. Recall that the intersection of two sets W1 and W2 is the set of all elements common to both sets, and the union of W1 and W2 is the set of elements that are in either W1 or W2. Suppose W1 and W2 are subspaces of a vector space V. Prove, or disprove by counterexample, the following propositions:
    1. W1 ∩ W2 is a subspace of V.
    2. W1 ∪ W2 is a subspace of V.