Chapter 3: Euclidean Vector Spaces
Multiple Choice Questions
1. Let u = 1, 0, −2 and v = 3, 2, 4 . Compute 2u + v.
(A) 4, 2, 2
(B) 5, 2, 0
(C) 6, 5, 0
(D) 7, 2, 2
2. What is the terminal point of a vector with initial point A −1, 0 that is equivalent to
u = 3, 4 ?
(A) 3, 3
(B) 4, 4
(C) 3, 4
(D) 2, 4
3. Let kvk = 3. Which of the following vectors is a possible value of v?
(A) 1, 4, 0
(B) −1, −2, 0
(C) 3, 3, 3
(D) 2, −2, 1
4. If u, v, and w are vectors in Rn , which of the following expressions does not make
mathematical sense?
(A) (u + v) · w
(B) kuk · (v · w)
(C) u · v + kwk
(D) k ku − vk
5. Which of the following pairs of vectors are orthogonal?
(A) u = 1, 1, −1 , v = 2, 2, 2
(B) u = 1, 0, −1 , v = 2, 0, 1
(C) u = 6, 1, −5 , v = 2, 3, 3
(D) u = 6, −1, −5 , v = 2, 3, 3
Elementary Linear Algebra 11e –2– Anton/Rorres
6. The point-normal equation 2(x + 1) + y + (z − 2) = 0 goes through the point P and is
normal to n. What are P and n?
(A) P −1, 0, 2 , n = 2, 1, 1
(B) P 2, 1, 1 , n = −1, 0, 2
(C) P −2, −1, −1 , n = −1, 0, 2
(D) P 1, 0, −2 , n = 2, 1, 1
7. Which of the following is the
correct pair of vector and parametric
equations for the line
containing the point P 4, 0, 2 and parallel to v = 1, −4, 2 ?
(A) Parametric: x = 4 + t, y = −4t, z = 2 + 2t
Vector: x, y, z = 4, 0, 2 + t 1, −4, 2
(B) Parametric: x = 1 + 4t, y = −4,
z = 2 + 2t
Vector: x, y, z = t 1, −4, 2
(C) Parametric: x = t, y = −4t, z =
2t
Vector: x, y, z = t 1, −4, 2
(D) Parametric: x = 1, y = −4,z = 2
Vector: x, y, z = 4, 0, 2 + t 1, −4, 2
8. Given the equation of the line x = 8 − t, 6 + 2t , which of the following points is not on
the line?
(A) 6, 10 (B) 7, 9 (C) 3, 16 (D) 8, 6
9. If u, v, and w are vectors in Rn , which of the following expressions does not make
mathematical sense?
(A) (u + v) × w
(B) ku × vk (w · z)
(C) ku × vk × w
(D) (u × v) · w
10. What is the area of the triangle defined by the points 1, 3, 2 , 2, 3, 1 , and 2, 2, 3 ?
√ √
(A) 21 6 (B) 6 (C) 0 (D) 12
Free Response Questions
−→
1. Find the terminal point Q of a nonzero vector u = P Q with initial point P 4, −3, 6
such that u is oppositely directed to v = 1, 1, 7 .
2. Let P be the point 1, 3, 0 and Q be the point 5, 9, 4 . Find the midpoint of the line
segment between these two points.
3. Find a, b and c if 4, b, c = a 2, 3, 2 .
Chapter 3
Elementary Linear Algebra 11e –3– Anton/Rorres
4. Find the scalars c1 , c2 , and c3 for which the following equation is satisfied.
c1 1, 1, 1 + c2 1, 1, 0 + c3 1, 0, 0 = 2, −3, 4
5. Let u = 1, 0, −2 , v = 1, 1, 1 , and w = 2, −2, 1 . Calculate ku + vk − 2 kwk.
6. Find the angle between the vectors 2, 3, 1 and 4, 1, 2 .
7. If u = 1, k, −2, 5 and kuk = 6, what is the value of k?
8. Prove that if u and v are unit vectors, then −1 ≤ u · v ≤ 1.
9. Prove using the definition of the norm that for vectors u and v in Rn , ku − vk = kv − uk.
10. Calculate the distance between the point 2, 1 and the line x − 2y + 3 = 0.
11. Calculate the distance between the point 3, 1, −2 and the plane 2x + y − z + 1 = 0.
12. Let u, v, and w be vectors in Rn . Prove that if u is orthogonal to both v and w, then
u is orthogonal to the vector c1 v + c2 w for any scalars c1 and c2 .
13. Find proja u where a = 2, 5, 4 and u = 1, −2, 3 .
14. Find an equation for the plane containing the point 1, 1, 3 that is perpendicular to
the line x = 2 − 3t, y = 1 + t, z = 2t.
15. Find a vector perpendicular to the line x, y, z = 1, 3, 2 + t 0, 1, −4 .
16. Find the vector equation of the line which contains the point 1, −5, 2 and is perpen-
dicular to the plane 3x − 7y + 4z − 5 = 0.
17. Consider the system Ax = b. Prove that if x1 is a solution to Ax = b and x1 − x2 is a
solution to the corresponding homogeneous system, then x2 is also a solution to Ax = b.
18. Let u = 1, 1, 3 , v = 2, 0, 1 , and w = 0, −1, 5 . Compute ku × (v + w)k.
19. Calculate
the volume of the parallelepipeddefined by the vectors
u = 2, 0, 5 , v = 1, −1, 0 , and w = 1, 3, 8 .
20. A plane contains the vectors u = 1, 3, −4 and v = 4, 1, −2 . Find a vector normal
to the plane.
21. Let u and v be vectors in R3 . Prove that ku × vk2 = (u · u)(v · v) − (u · v)2 .
Chapter 3
Elementary Linear Algebra 11e –4– Anton/Rorres
Answers
Multiple Choice Answers
1. (B)
2. (D)
3. (D)
4. (B)
5. (C)
6. (A)
7. (A)
8. (B)
9. (C)
10. (A)
Free Response Answers
1. Possible answer: Q = 3, −4, −1
2. 3, 6, 2
3. a = 2, b = 6, c = 4
4. c1 = 4, c2 = −7, c3 = 5
√
5. 6−6
6. 40.696◦
√
7. k = ± 6
10. √3
5
10
11. √
6
Chapter 3
Elementary Linear Algebra 11e –5– Anton/Rorres
8
, 4 , 16
13. 45 9 45
14. −3(x − 1) + (y − 1) + 2(z − 3) = 0
15. Possible answer: 0, 4, 1
16. x = 1, −5, 2 + t 3, −7, 4
√
18. 3 10
19. 4
20. Possible answer: −2, −14, −11 or 2, 14, 11
Chapter 3
