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Dot Product - Calculus - Exercise, Exercises of Calculus

National Institute of Industrial EngineeringCalculus

This file contains some problems related calculus. Some hints to the given problems are: Dot Product, Vectors, Dot Product, Find the Angle, Given Vectors, Nearest Tenth, Degree, Two Orthogonal Vectors, Vector Projection, Minimum Force

Typology: Exercises

2011/2012

Uploaded on 12/31/2012

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aparijita 🇮🇳

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bg1
Vectors and Dot Product
Finda·b.
1) a=2
,
4
,
b=2
,
5
A) 4
,
20 B) 4
,
9 C) 16 D) 24
2) a=3
,
-1
,
b=-3
,
1
A) -9
,
-1 B) 0
,
0 C) -10 D) 8
3) a=5
,
-10
,
b=6
,
5
A) 11
,
-5 B) 80 C) 30
,
-50 D) -20
4) a=4
,
-6
,
b=5
,
-7
A) 62 B) 22 C) 9
,
-13 D) 20
,
42
5) a=-1
,
-1
,
b=-3.5
,
2
A) -4.5
,
1 B) 5.5 C) 3.5
,
-2 D) 1.5
6) a=7i-4j
,
b=4i+3j
A) 11
,
-1 B) -40 C) 16 D) 28
,
-12
7) a=4i+8j
,
b=-2i+3j
A) -8
,
24 B) 2
,
11 C) 32 D) 16
8) a=6j
,
b=i+4j
A) 0,24 B) 1,24 C) 24 D) -24
Usethedotproducttofindv.
9) v=1
,
7
A) 8 B) 4 3 C) 6 D) 5 2
10) v=-7
,
3
A) 58 B) 10 C) 2i 10 D) -4
11) v=5j
A) 25 B) 5 C) -5D)0
12) v=6
,
8
A) 14 B) 10 C) -2 D) 100
Findtheanglebetweenthegivenvectorstothenearesttenthofadegree.
13) u=-5
,
8
,
v=-4
,
8
A) -7.3° B) 2.7° C) 5.4° D) 15.4°
14) u=-2
,
-1
,
v=5
,
-9
A) 102.5° B) 46.3° C) 92.5° D) 36.3°
pf3
pf4

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Vectors and Dot Product

Find a· b.

  1. a = 2, 4 , b = 2, 5 A) 4, 20 B) 4, 9 C) 16 D) 24

  2. a = 3, - 1 , b = - 3, 1 A) - 9, - 1 B) 0, 0 C) - 10 D) 8

  3. a = 5, - 10 , b = 6, 5 A) 11, - 5 B) 80 C) 30, - 50 D) - 20

  4. a = 4, - 6 , b = 5, - 7 A) 62 B) 22 C) 9, - 13 D) 20, 42

  5. a = - 1, - 1 , b = - 3.5, 2 A) - 4.5, 1 B) 5.5 C) 3.5, - 2 D) 1.

  6. a = 7 i - 4 j , b = 4 i + 3 j A) 11, - 1 B) - 40 C) 16 D) 28, - 12

  7. a = 4 i + 8 j , b = - 2 i + 3 j A) - 8, 24 B) 2, 11 C) 32 D) 16

  8. a = 6 j , b = i + 4 j A) 0, 24 B) 1, 24 C) 24 D) - 24

Use the dot product to find v.

  1. v = 1, 7 A) 8 B) 4 3 C) 6 D) 5 2

  2. v = - 7, 3 A) 58 B) 10 C) 2i 10 D) - 4

  3. v = 5 j A) 25 B) 5 C) - 5 D) 0

  4. v = 6, 8 A) 14 B) 10 C) - 2 D) 100

Find the angle between the given vectors to the nearest tenth of a degree.

  1. u = - 5, 8 , v = - 4, 8 A) - 7.3° B) 2.7° C) 5.4° D) 15.4°

  2. u = - 2, - 1 , v = 5, - 9 A) 102.5° B) 46.3° C) 92.5° D) 36.3°

PreCalculus

  1. u = 5 i - 9 j , v = 5 i + j A) 65.9° B) 90.4° C) 100.1° D) 45.2°

  2. u = i + 7 j , v = - i + 8 j A) 13.6° B) 68.9° C) 27.8° D) 15.3°

  3. u = 4, - 3 , v = - 6, - 8 A) 90° B) 45° C) 180° D) 0°

  4. u = 5 cos π 6

i + 5 sin π 6

j, v = cos 4 π 3

i + sin 4 π 3

j

A) 120° B) 150° C) 210° D) 240°

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, projvu.

  1. u = 8, 6 , v = 2, 4 A) 2, 4 + 6, 2 B) 3, 6 + 5, - 2.5 C) 4, 8 + 4, - 2 D) - 1, - 2 + 9, - 4.

  2. u = 5, - 5 , v = 6, 8 A) 3, 4 + - 8, 6 B) - 0.6, - 0.8 + 5.6, - 4. C) 1.5, 2 + 3.5, - 7 D) 6, 8 + -1, - 13

  3. u = - 3, - 3 , v = - 9, - 3 A) 3, 1 + - 6, - 4 B) - 6, - 2 + 3, - 9 C) - 3.6, - 1.2 + 0.6, - 1.8 D) - 9, - 3 + 6, 0

Determine whether the vectors u and v are parallel, orthogonal, or neither.

  1. u = 10, 0 , v = 0, - 9 A) Orthogonal B) Neither C) Parallel

  2. u = 3, 0 , v = 0, - 3 A) Neither B) Parallel C) Orthogonal

  3. u = 6, - 2 , v = 8, 24 A) Parallel B) Orthogonal C) Neither

  4. u = 7, 2 , v = 21, 6 A) Neither B) Parallel C) Orthogonal

  5. u = 8, 4 , v = 10, 7 A) Parallel B) Neither C) Orthogonal

Solve the problem.

  1. What is the minimum force required to prevent a ball weighing 24.0 lb from rolling down a ramp which is inclined 27.2° with the horizontal? A) 11 lb B) 10.7 lb C) 5.5 lb D) 21.3 lb

Answer Key

Testname: 4_DOT_PRODUCT

1) D

2) C

3) D

4) A

5) D

6) C

7) D

8) C

9) D

10) A

11) B

12) B

13) C

14) C

15) C

16) C

17) A

18) B

19) C

20) B

21) C

22) A

23) C

24) B

25) B

26) B

27) A

28) D

29) A

30) C