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mf mf mf mf mf Page 1
mf
.
, Test Bank For mf mf
Linear Algebra A Modern Introduction 5th Edition by David Poole
mf mf mf mf mf mf mf mf mf mf
Section 1.0 - 1.4 mf mf mf
1. If u • v = 0, then ||u + v|| = ||u – v||.
mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
2. If u • v = u • w, then either u = 0 or v = w.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
3. a • b × c = 0 if and only if the vectors a, b, c are coplanar.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
n
located by the vectors u and v is ||u – v||.
mf
4. The distance between two points in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
5. If v is any nonzero vector, then 6v is a vector in the same direction as v with a length of 6 units.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
6. The only real number c for which [c, –2, 1] is orthogonal to [2c, c, –4] is c = 2.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
7. The projection of a vector v onto a vector u is undefined if v = 0.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
8. The area of the parallelogram with sides a, b, is
mf mf mf mf mf mf mf mf mf m f mf || ||
mf
a. True
b. False
2 2 2 2
, then (a × b • c) = ||a|| ||b|| ||c|| .
mf
9. If a, b, c are mutually orthogonal vectors in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
10. For all vectors v and scalars c, ||cv|| = c||v||.
mf mf mf mf mf mf mf mf mf
a. True
b. False
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mf mf mf mf mf Page 2
mf
.
, n
11. For all vectors u, v, w in
mf mf mf mf mf mf mf , u – (v – w) = u + w – v.
mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
12. The projection of a vector v onto a vector u is undefined if u = 0.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
13. The vectors [1, 2, 3] and [k, 2k, 3k] have the same direction for all nonzero real numbers k?
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
14. If a parity check code is used in the transmission of a message consisting of a binary vector, then the total number
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
of 1’s in the message will be even.
mf mf mf mf mf mf mf
a. True
b. False
15. The distance between the planes n • x = d1 and n • x = d2 is |d1 – d2|.
mf mf mf mf mf mf mf mf mf
mf
mf mf mf mf mf
mf
mf
mf
mf
a. True
b. False
16. The zero vector is orthogonal to every vector except itself.
mf mf mf mf mf mf mf mf mf
a. True
b. False
17. The products a × (b × c) and (a × b) × c are equal if and only if b = 0.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
18. Simplify the following vector expression: 4u – 2(v + 3w) + 6(w
mf mf mf mf mf mf mf mf mf mf mf mf mf u).
19. Find all solutions of 3x + 5 = 2 in
mf mf mf mf mf mf mf mf mf mf , or show that there are no solutions.
mf mf mf mf mf mf mf
a. 2
m f
b. 4
m f
c. 6
m f
d. 8
m f
Find the distance between the parallel lines.
mf mf mf mf mf mf mf
20.
and
mf mf
21. Find the acute angle between the planes
mf mf mf mf mf mf mf mf 3 and mf mf .
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mf mf mf mf mf Page 3
mf
.
, 22. Find the distance between the planes mf mf mf mf mf mf m f and mf .
23. Find values of the scalar k for which the following vectors are orthogonal.
mf mf mf mf mf mf mf mf mf mf mf mf
u = [k, k, –2], v = [–2, k – 1, 5]
mf mf mf mf mf mf mf mf mf mf mf
24. Simplify the following expressions: mf mf mf
(a) (a + b + c) × c + (a + b + c) × b + (b – c) × a
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
(b) (v + 2w) ∙ (w + z) × (3z + v)
mf mf mf mf mf mf mf mf mf mf
25. Find the check digit that should be appended to the vector u = [2, 5, 6, 4, 5] in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf if the check vector is c = [1, 1, 1, 1,
mf mf mf mf mf mf mf mf mf mf
1, 1]. mf
26. If u is orthogonal to v, then which of the following is also orthogonal to v?
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
27. What is the distance of the point P = (2, 3, –1) to the line of intersection of the planes 2x – 2y + z = –3 and 3x –
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
2y + 2z = –17?
mf mf mf mf mf
28. In a parallelogram ABCD let
mf mf mf mf mf mf = a, mf mf mf b. Let M be the point of intersection of the diagonals. Express
mf mf mf mf mf mf mf mf mf mf mf mf
, mf mf and m f m f as linear combinations of a and b.
mf mf mf mf mf mf
29. Suppose that the dot product of u = [u1, u2] and v = [v1, v2] in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
2 mf
were defined as u · v = 5u1 v1 + 2u2 v2. Consider the following statements for vectors u, v, w, and all scalars c.
mf mf mf mf mf mf mf
mf mf
mf
mf
mf mf mf mf mf mf mf mf mf mf mf mf mf
a. u · v = v · umf mf mf mf mf mf
b. u · (v + w) = u · v + u · w
mf mf mf mf mf mf mf mf mf mf mf mf
c. (cu) · v = c(u · v) mf mf mf mf mf mf
d. u · u ≥ 0 and u · u = 0 if and inly if u = 0
mf mf mf mf mf m f mf mf mf mf mf mf mf mf mf mf mf
30. Find a value of k so that the angle between the line 4x + ky = 20 and the line 2x – 3y = –6 is 45°.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
31. Find the orthogonal projection of v = [–1, 2, 1] onto the xz-plane.
mf mf mf mf mf mf mf mf mf mf mf mf
32. Show that the quadrilateral with vertices A = (–3, 5, 6), B = (1, –5, 7), C = (8, –3, –1) and D = (4, 7, –2) is a square.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
33. If a = [1, –2, 3], b = [4, 0, 1], c = [2, 1, –3], compute 2a – 3b + 4c.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
3
that is perpendicular to the plane 2x – 3y + 7z –
mf
34. Find the vector parametric equation of the line in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
4 = 0 and which passes through the point P = (l, –5, 7).
mf mf mf mf mf mf mf mf mf mf mf mf mf mf
35. Find all values of k such that d(a, b) = 6, where a = [2, k, 1, –4] and b = [3, –1, 6, –3].
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
36. Show that if a vector v is orthogonal to two noncollinear vectors in a plane P, then v is orthogonal to every vector in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
P.
37. Final all solutions of 7x = 1 in mf mf mf mf mf mf mf mf , or show that there are no solutions.
mf mf mf mf mf mf mf
38. Let u1 and u2 be unit vectors, and let the angle between them be
mf
mf
mf
mf
mf mf mf mf mf mf mf mf mf m f
m f radians. What is the area of the parallelogram whose diagonals are d1 = 2u1 – u2 and d2 = 4u1 –5u2?
mf mf mf mf mf mf mf mf mf mf mf
mf
mf
mf
mf
mf
mf
mf
mf
mf
Copyright Cengage Learning. Powered by Cognero
mf mf mf mf mf Page 4mf
.
mf mf mf mf mf Page 1
mf
.
, Test Bank For mf mf
Linear Algebra A Modern Introduction 5th Edition by David Poole
mf mf mf mf mf mf mf mf mf mf
Section 1.0 - 1.4 mf mf mf
1. If u • v = 0, then ||u + v|| = ||u – v||.
mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
2. If u • v = u • w, then either u = 0 or v = w.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
3. a • b × c = 0 if and only if the vectors a, b, c are coplanar.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
n
located by the vectors u and v is ||u – v||.
mf
4. The distance between two points in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
5. If v is any nonzero vector, then 6v is a vector in the same direction as v with a length of 6 units.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
6. The only real number c for which [c, –2, 1] is orthogonal to [2c, c, –4] is c = 2.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
7. The projection of a vector v onto a vector u is undefined if v = 0.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
8. The area of the parallelogram with sides a, b, is
mf mf mf mf mf mf mf mf mf m f mf || ||
mf
a. True
b. False
2 2 2 2
, then (a × b • c) = ||a|| ||b|| ||c|| .
mf
9. If a, b, c are mutually orthogonal vectors in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
10. For all vectors v and scalars c, ||cv|| = c||v||.
mf mf mf mf mf mf mf mf mf
a. True
b. False
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mf mf mf mf mf Page 2
mf
.
, n
11. For all vectors u, v, w in
mf mf mf mf mf mf mf , u – (v – w) = u + w – v.
mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
12. The projection of a vector v onto a vector u is undefined if u = 0.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
13. The vectors [1, 2, 3] and [k, 2k, 3k] have the same direction for all nonzero real numbers k?
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
14. If a parity check code is used in the transmission of a message consisting of a binary vector, then the total number
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
of 1’s in the message will be even.
mf mf mf mf mf mf mf
a. True
b. False
15. The distance between the planes n • x = d1 and n • x = d2 is |d1 – d2|.
mf mf mf mf mf mf mf mf mf
mf
mf mf mf mf mf
mf
mf
mf
mf
a. True
b. False
16. The zero vector is orthogonal to every vector except itself.
mf mf mf mf mf mf mf mf mf
a. True
b. False
17. The products a × (b × c) and (a × b) × c are equal if and only if b = 0.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
a. True
b. False
18. Simplify the following vector expression: 4u – 2(v + 3w) + 6(w
mf mf mf mf mf mf mf mf mf mf mf mf mf u).
19. Find all solutions of 3x + 5 = 2 in
mf mf mf mf mf mf mf mf mf mf , or show that there are no solutions.
mf mf mf mf mf mf mf
a. 2
m f
b. 4
m f
c. 6
m f
d. 8
m f
Find the distance between the parallel lines.
mf mf mf mf mf mf mf
20.
and
mf mf
21. Find the acute angle between the planes
mf mf mf mf mf mf mf mf 3 and mf mf .
Copyright Cengage Learning. Powered by Cognero
mf mf mf mf mf Page 3
mf
.
, 22. Find the distance between the planes mf mf mf mf mf mf m f and mf .
23. Find values of the scalar k for which the following vectors are orthogonal.
mf mf mf mf mf mf mf mf mf mf mf mf
u = [k, k, –2], v = [–2, k – 1, 5]
mf mf mf mf mf mf mf mf mf mf mf
24. Simplify the following expressions: mf mf mf
(a) (a + b + c) × c + (a + b + c) × b + (b – c) × a
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
(b) (v + 2w) ∙ (w + z) × (3z + v)
mf mf mf mf mf mf mf mf mf mf
25. Find the check digit that should be appended to the vector u = [2, 5, 6, 4, 5] in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf if the check vector is c = [1, 1, 1, 1,
mf mf mf mf mf mf mf mf mf mf
1, 1]. mf
26. If u is orthogonal to v, then which of the following is also orthogonal to v?
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
27. What is the distance of the point P = (2, 3, –1) to the line of intersection of the planes 2x – 2y + z = –3 and 3x –
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
2y + 2z = –17?
mf mf mf mf mf
28. In a parallelogram ABCD let
mf mf mf mf mf mf = a, mf mf mf b. Let M be the point of intersection of the diagonals. Express
mf mf mf mf mf mf mf mf mf mf mf mf
, mf mf and m f m f as linear combinations of a and b.
mf mf mf mf mf mf
29. Suppose that the dot product of u = [u1, u2] and v = [v1, v2] in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
2 mf
were defined as u · v = 5u1 v1 + 2u2 v2. Consider the following statements for vectors u, v, w, and all scalars c.
mf mf mf mf mf mf mf
mf mf
mf
mf
mf mf mf mf mf mf mf mf mf mf mf mf mf
a. u · v = v · umf mf mf mf mf mf
b. u · (v + w) = u · v + u · w
mf mf mf mf mf mf mf mf mf mf mf mf
c. (cu) · v = c(u · v) mf mf mf mf mf mf
d. u · u ≥ 0 and u · u = 0 if and inly if u = 0
mf mf mf mf mf m f mf mf mf mf mf mf mf mf mf mf mf
30. Find a value of k so that the angle between the line 4x + ky = 20 and the line 2x – 3y = –6 is 45°.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
31. Find the orthogonal projection of v = [–1, 2, 1] onto the xz-plane.
mf mf mf mf mf mf mf mf mf mf mf mf
32. Show that the quadrilateral with vertices A = (–3, 5, 6), B = (1, –5, 7), C = (8, –3, –1) and D = (4, 7, –2) is a square.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
33. If a = [1, –2, 3], b = [4, 0, 1], c = [2, 1, –3], compute 2a – 3b + 4c.
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
3
that is perpendicular to the plane 2x – 3y + 7z –
mf
34. Find the vector parametric equation of the line in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
4 = 0 and which passes through the point P = (l, –5, 7).
mf mf mf mf mf mf mf mf mf mf mf mf mf mf
35. Find all values of k such that d(a, b) = 6, where a = [2, k, 1, –4] and b = [3, –1, 6, –3].
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
36. Show that if a vector v is orthogonal to two noncollinear vectors in a plane P, then v is orthogonal to every vector in
mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf mf
P.
37. Final all solutions of 7x = 1 in mf mf mf mf mf mf mf mf , or show that there are no solutions.
mf mf mf mf mf mf mf
38. Let u1 and u2 be unit vectors, and let the angle between them be
mf
mf
mf
mf
mf mf mf mf mf mf mf mf mf m f
m f radians. What is the area of the parallelogram whose diagonals are d1 = 2u1 – u2 and d2 = 4u1 –5u2?
mf mf mf mf mf mf mf mf mf mf mf
mf
mf
mf
mf
mf
mf
mf
mf
mf
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mf mf mf mf mf Page 4mf
.
