Copyright Cengage Learning. Powered by Cognero.
st st st st st Page 1
st
, Test Bank For st st
Linear Algebra A Modern Introduction 5th Edition by David Poole Copyright 2026
st st st st st st st st st st st
Section 1.0 - 1.4 st st st
1. If u • v = 0, then ||u + v|| = ||u – v||.
st st st st st st st st st st st st st
a. True
b. False
2. If u • v = u • w, then either u = 0 or v = w.
st st st st st st st st st st st st st st st st
a. True
b. False
3. a • b × c = 0 if and only if the vectors a, b, c are coplanar.
st st st st st st st st st st st st st st st st st
a. True
b. False
n
located by the vectors u and v is ||u – v||.
st
4. The distance between two points in
st st st st st st st st st st st st st st st st
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.
st st st st st st st st st st st st st st st st st st st st st st
a. True
b. False
6. The only real number c for which [c, –2, 1] is orthogonal to [2c, c, –4] is c = 2.
st st st st st st st st st st st st st st st st st st st
a. True
b. False
7. The projection of a vector v onto a vector u is undefined if v = 0.
st st st st st st st st st st st st st st st
a. True
b. False
8. The area of the parallelogram with sides a, b, is
st st st st st st st st st s t st || st ||
a. True
b. False
2 2 2 2
, then (a × b • c) = ||a|| ||b|| ||c|| .
st
9. If a, b, c are mutually orthogonal vectors in
st st st st st st st st st st st st st st st st
a. True
b. False
10. For all vectors v and scalars c, ||cv|| = c||v||.
st st st st st st st st st
a. True
b. False
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st st st st st Page 2
st
, n
11. For all vectors u, v, w in
st st st st st st st , u – (v – w) = u + w – v.
st st st st st st st st st st st
a. True
b. False
12. The projection of a vector v onto a vector u is undefined if u = 0.
st st st st st st st st st st st st st st st
a. True
b. False
13. The vectors [1, 2, 3] and [k, 2k, 3k] have the same direction for all nonzero real numbers k?
st st st st st st st st st st st st st st st st st st
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 of 1’s i
st st st st st st st st st st st st st st st st st st st st st st st st
n the message will be even.
st st st st st
a. True
b. False
15. The distance between the planes n • x = d1 and n • x = d2 is |d1 – d2|.
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st
st st st st st
st
st
st
st
a. True
b. False
16. The zero vector is orthogonal to every vector except itself.
st st st st st st st st st
a. True
b. False
17. The products a × (b × c) and (a × b) × c are equal if and only if b = 0.
st st st st st st st st st st st st st st st st st st st st st
a. True
b. False
18. Simplify the following vector expression: 4u – 2(v + 3w) + 6(w
st st st st st st st st st st st st st u).
19. Find all solutions of 3x + 5 = 2 in
st st st st st st st st st st , or show that there are no solutions.
st st st st st st st
a. 2 s t
b. 4 s t
c. 6 s t
d. 8 s t
Find the distance between the parallel lines.
st st st st st st st
20.
and
st st
21. Find the acute angle between the planes
st st st st st st st st 3 and
st st .
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st st st st st Page 3
st
, 22. Find the distance between the planes st st st st st st s t and st .
23. Find values of the scalar k for which the following vectors are orthogonal.
st st st st st st st st st st st st
u = [k, k, –2], v = [–2, k – 1, 5]
st st st st st st st st st st st
24. Simplify the following expressions: st st st
(a) (a + b + c) × c + (a + b + c) × b + (b – c) × a
st st st st st st st st st st st st st st st st st st st st
(b) (v + 2w) ∙ (w + z) × (3z + v)
st st st st st st st st st st
25. Find the check digit that should be appended to the vector u = [2, 5, 6, 4, 5] in
st st st st st st st st st st st st st st st st st st st st if the check vector is c = [1, 1, 1, 1,
st st st st st st st st st st
1, 1]. st
26. If u is orthogonal to v, then which of the following is also orthogonal to v?
st st st st st st st st st st st st st st st
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 – 2y + 2z = –
st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st
17?
28. In a parallelogram ABCD let st st st st st st = a, st st st b. Let M be the point of intersection of the diagonals. Express
st st st st st st st st st st st st , st
st and s t s t as linear combinations of a and b.
st st st st st st
29. Suppose that the dot product of u = [u1, u2] and v = [v1, v2] in st st st st st st st st st st st st st st st st
2 st
were defined as u · v = 5u1 v1 + 2u2 v2. Consider the following statements for vectors u, v, w, and all scalars c.
st st st st st st st
st st
st
st
st st st st st st st st st st st st st
a. u · v = v · u st st st st st st
b. u · (v + w) = u · v + u · w
st st st st st st st st st st st st
c. (cu) · v = c(u · v) st st st st st st
d. u · u ≥ 0 and u · u = 0 if and inly if u = 0
st st st st st s t st st st st st st st st st st st
30. Find a value of k so that the angle between the line 4x + ky = 20 and the line 2x – 3y = –6 is 45°.
st st st st st st st st st st st st st st st st st st st st st st st st st st
31. Find the orthogonal projection of v = [–1, 2, 1] onto the xz-plane.
st st st st st st st st st st st st
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.
st st st st st st st st st st st st st st st st st st st st st st st st st st st st st
33. If a = [1, –2, 3], b = [4, 0, 1], c = [2, 1, –3], compute 2a – 3b + 4c.
st st st st st st st st st st st st st st st st st st st st st
3
34. Find the vector parametric equation of the line in that is perpendicular to the plane 2x – 3y + 7z –
st
st st st st st st st st st st st st st st st st st st st st
4 = 0 and which passes through the point P = (l, –5, 7).
st st st st st st st st st st st st st st
35. Find all values of k such that d(a, b) = 6, where a = [2, k, 1, –4] and b = [3, –1, 6, –3].
st st st st st st st st st st st st st st st st st st st st st st st st
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
st st st st st st st st st st st st st st st st st st st st st st st
P.
37. Final all solutions of 7x = 1 in st st st st st st st st , or show that there are no solutions.
st st st st st st st
38. Let u1 and u2 be unit vectors, and let the angle between them be
st
st
st
st
st st st st st st st st st s t
s t radians. What is the area of the parallelogram whose diagonals are d1 = 2u1 – u2 and d2 = 4u1 –5u2?
st st st st st st st st st st st
st
st
st
st
st
st
st
st
st
Copyright Cengage Learning. Powered by Cognero. st st st st st Page 4 st
st st st st st Page 1
st
, Test Bank For st st
Linear Algebra A Modern Introduction 5th Edition by David Poole Copyright 2026
st st st st st st st st st st st
Section 1.0 - 1.4 st st st
1. If u • v = 0, then ||u + v|| = ||u – v||.
st st st st st st st st st st st st st
a. True
b. False
2. If u • v = u • w, then either u = 0 or v = w.
st st st st st st st st st st st st st st st st
a. True
b. False
3. a • b × c = 0 if and only if the vectors a, b, c are coplanar.
st st st st st st st st st st st st st st st st st
a. True
b. False
n
located by the vectors u and v is ||u – v||.
st
4. The distance between two points in
st st st st st st st st st st st st st st st st
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.
st st st st st st st st st st st st st st st st st st st st st st
a. True
b. False
6. The only real number c for which [c, –2, 1] is orthogonal to [2c, c, –4] is c = 2.
st st st st st st st st st st st st st st st st st st st
a. True
b. False
7. The projection of a vector v onto a vector u is undefined if v = 0.
st st st st st st st st st st st st st st st
a. True
b. False
8. The area of the parallelogram with sides a, b, is
st st st st st st st st st s t st || st ||
a. True
b. False
2 2 2 2
, then (a × b • c) = ||a|| ||b|| ||c|| .
st
9. If a, b, c are mutually orthogonal vectors in
st st st st st st st st st st st st st st st st
a. True
b. False
10. For all vectors v and scalars c, ||cv|| = c||v||.
st st st st st st st st st
a. True
b. False
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st st st st st Page 2
st
, n
11. For all vectors u, v, w in
st st st st st st st , u – (v – w) = u + w – v.
st st st st st st st st st st st
a. True
b. False
12. The projection of a vector v onto a vector u is undefined if u = 0.
st st st st st st st st st st st st st st st
a. True
b. False
13. The vectors [1, 2, 3] and [k, 2k, 3k] have the same direction for all nonzero real numbers k?
st st st st st st st st st st st st st st st st st st
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 of 1’s i
st st st st st st st st st st st st st st st st st st st st st st st st
n the message will be even.
st st st st st
a. True
b. False
15. The distance between the planes n • x = d1 and n • x = d2 is |d1 – d2|.
st st st st st st st st st
st
st st st st st
st
st
st
st
a. True
b. False
16. The zero vector is orthogonal to every vector except itself.
st st st st st st st st st
a. True
b. False
17. The products a × (b × c) and (a × b) × c are equal if and only if b = 0.
st st st st st st st st st st st st st st st st st st st st st
a. True
b. False
18. Simplify the following vector expression: 4u – 2(v + 3w) + 6(w
st st st st st st st st st st st st st u).
19. Find all solutions of 3x + 5 = 2 in
st st st st st st st st st st , or show that there are no solutions.
st st st st st st st
a. 2 s t
b. 4 s t
c. 6 s t
d. 8 s t
Find the distance between the parallel lines.
st st st st st st st
20.
and
st st
21. Find the acute angle between the planes
st st st st st st st st 3 and
st st .
Copyright Cengage Learning. Powered by Cognero.
st st st st st Page 3
st
, 22. Find the distance between the planes st st st st st st s t and st .
23. Find values of the scalar k for which the following vectors are orthogonal.
st st st st st st st st st st st st
u = [k, k, –2], v = [–2, k – 1, 5]
st st st st st st st st st st st
24. Simplify the following expressions: st st st
(a) (a + b + c) × c + (a + b + c) × b + (b – c) × a
st st st st st st st st st st st st st st st st st st st st
(b) (v + 2w) ∙ (w + z) × (3z + v)
st st st st st st st st st st
25. Find the check digit that should be appended to the vector u = [2, 5, 6, 4, 5] in
st st st st st st st st st st st st st st st st st st st st if the check vector is c = [1, 1, 1, 1,
st st st st st st st st st st
1, 1]. st
26. If u is orthogonal to v, then which of the following is also orthogonal to v?
st st st st st st st st st st st st st st st
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 – 2y + 2z = –
st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st st
17?
28. In a parallelogram ABCD let st st st st st st = a, st st st b. Let M be the point of intersection of the diagonals. Express
st st st st st st st st st st st st , st
st and s t s t as linear combinations of a and b.
st st st st st st
29. Suppose that the dot product of u = [u1, u2] and v = [v1, v2] in st st st st st st st st st st st st st st st st
2 st
were defined as u · v = 5u1 v1 + 2u2 v2. Consider the following statements for vectors u, v, w, and all scalars c.
st st st st st st st
st st
st
st
st st st st st st st st st st st st st
a. u · v = v · u st st st st st st
b. u · (v + w) = u · v + u · w
st st st st st st st st st st st st
c. (cu) · v = c(u · v) st st st st st st
d. u · u ≥ 0 and u · u = 0 if and inly if u = 0
st st st st st s t st st st st st st st st st st st
30. Find a value of k so that the angle between the line 4x + ky = 20 and the line 2x – 3y = –6 is 45°.
st st st st st st st st st st st st st st st st st st st st st st st st st st
31. Find the orthogonal projection of v = [–1, 2, 1] onto the xz-plane.
st st st st st st st st st st st st
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.
st st st st st st st st st st st st st st st st st st st st st st st st st st st st st
33. If a = [1, –2, 3], b = [4, 0, 1], c = [2, 1, –3], compute 2a – 3b + 4c.
st st st st st st st st st st st st st st st st st st st st st
3
34. Find the vector parametric equation of the line in that is perpendicular to the plane 2x – 3y + 7z –
st
st st st st st st st st st st st st st st st st st st st st
4 = 0 and which passes through the point P = (l, –5, 7).
st st st st st st st st st st st st st st
35. Find all values of k such that d(a, b) = 6, where a = [2, k, 1, –4] and b = [3, –1, 6, –3].
st st st st st st st st st st st st st st st st st st st st st st st st
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
st st st st st st st st st st st st st st st st st st st st st st st
P.
37. Final all solutions of 7x = 1 in st st st st st st st st , or show that there are no solutions.
st st st st st st st
38. Let u1 and u2 be unit vectors, and let the angle between them be
st
st
st
st
st st st st st st st st st s t
s t radians. What is the area of the parallelogram whose diagonals are d1 = 2u1 – u2 and d2 = 4u1 –5u2?
st st st st st st st st st st st
st
st
st
st
st
st
st
st
st
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