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Dot product and angles practice problems

u · v, lengths, and the angle between two vectors. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 9.1

easy
Let \( \displaystyle \mathbf{u} = \langle 1, 3, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, -4, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 3 \left(-4\right) + 0 \left(-5\right) + 1 \cdot 3 = -9 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{10} \]
    |u|.Reviewed
  3. \[ 5 \sqrt{2} \]
    |v|.Reviewed
  4. \[ - \frac{9 \sqrt{5}}{50} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -9,\quad \theta = \operatorname{acos}{\left(- \frac{9 \sqrt{5}}{50} \right)} \approx 113.73^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.2

easy
Let \( \displaystyle \mathbf{u} = \langle -1, 3, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, 4, 0 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-1\right) 0 - 3 \cdot 0 + 3 \cdot 4 = 12 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{19} \]
    |u|.Reviewed
  3. \[ 4 \]
    |v|.Reviewed
  4. \[ \frac{3 \sqrt{19}}{19} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 12,\quad \theta = \operatorname{acos}{\left(\frac{3 \sqrt{19}}{19} \right)} \approx 46.51^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.3

easy
Let \( \displaystyle \mathbf{u} = \langle 0, -4, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 3, -4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-4\right) 3 + 0 \cdot 5 - 4 \left(-4\right) = 4 \]
    Multiply componentwise and add.✓ Proved
  2. \[ 4 \sqrt{2} \]
    |u|.Reviewed
  3. \[ 5 \sqrt{2} \]
    |v|.Reviewed
  4. \[ 1 \cdot \frac{1}{10} = \frac{1}{10} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 4,\quad \theta = \operatorname{acos}{\left(\frac{1}{10} \right)} \approx 84.26^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.4

easy
Let \( \displaystyle \mathbf{u} = \langle 4, -5, -2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, 1, -2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 4 \left(-4\right) - 5 \cdot 1 - 2 \left(-2\right) = -17 \]
    Multiply componentwise and add.✓ Proved
  2. \[ 3 \sqrt{5} \]
    |u|.Reviewed
  3. \[ \sqrt{21} \]
    |v|.Reviewed
  4. \[ - \frac{17 \sqrt{105}}{315} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -17,\quad \theta = \operatorname{acos}{\left(- \frac{17 \sqrt{105}}{315} \right)} \approx 123.57^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.6

easy
Let \( \displaystyle \mathbf{u} = \langle -1, 3, 4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, 2, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 4 \left(-5\right) - 1 \cdot 0 + 3 \cdot 2 = -14 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{26} \]
    |u|.Reviewed
  3. \[ \sqrt{29} \]
    |v|.Reviewed
  4. \[ - \frac{7 \sqrt{754}}{377} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -14,\quad \theta = \operatorname{acos}{\left(- \frac{7 \sqrt{754}}{377} \right)} \approx 120.65^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.7

easy
Let \( \displaystyle \mathbf{u} = \langle 4, 2, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, 2, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 4 \cdot 0 + 2 \cdot 2 + 1 \cdot 5 = 9 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{21} \]
    |u|.Reviewed
  3. \[ \sqrt{29} \]
    |v|.Reviewed
  4. \[ \frac{3 \sqrt{609}}{203} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 9,\quad \theta = \operatorname{acos}{\left(\frac{3 \sqrt{609}}{203} \right)} \approx 68.61^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.14

easy
Let \( \displaystyle \mathbf{u} = \langle -1, 4, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, -4, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 4 \left(-4\right) - 1 \cdot 3 + 2 \cdot 2 = -15 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{21} \]
    |u|.Reviewed
  3. \[ \sqrt{29} \]
    |v|.Reviewed
  4. \[ - \frac{5 \sqrt{609}}{203} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -15,\quad \theta = \operatorname{acos}{\left(- \frac{5 \sqrt{609}}{203} \right)} \approx 127.43^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.15

easy
Let \( \displaystyle \mathbf{u} = \langle -1, -5, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -2, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-1\right) 2 + 1 \cdot 4 - 5 \left(-2\right) = 12 \]
    Multiply componentwise and add.✓ Proved
  2. \[ 3 \sqrt{3} \]
    |u|.Reviewed
  3. \[ 2 \sqrt{6} \]
    |v|.Reviewed
  4. \[ \frac{\sqrt{2}}{3} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 12,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{2}}{3} \right)} \approx 61.87^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.46

easy
Let \( \displaystyle \mathbf{u} = \langle 0, 5, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 3, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 0 \left(-1\right) - 1 \left(-5\right) + 5 \cdot 3 = 20 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{26} \]
    |u|.Reviewed
  3. \[ \sqrt{35} \]
    |v|.Reviewed
  4. \[ \frac{2 \sqrt{910}}{91} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 20,\quad \theta = \operatorname{acos}{\left(\frac{2 \sqrt{910}}{91} \right)} \approx 48.47^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.47

easy
Let \( \displaystyle \mathbf{u} = \langle 4, 0, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 1, 0, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 0 \cdot 0 - 3 \left(-1\right) + 4 \cdot 1 = 7 \]
    Multiply componentwise and add.✓ Proved
  2. \[ 5 \]
    |u|.Reviewed
  3. \[ \sqrt{2} \]
    |v|.Reviewed
  4. \[ \frac{7 \sqrt{2}}{10} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 7,\quad \theta = \operatorname{acos}{\left(\frac{7 \sqrt{2}}{10} \right)} \approx 8.13^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.48

easy
Let \( \displaystyle \mathbf{u} = \langle 5, -1, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, 2, -4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 2 \left(-4\right) - 1 \cdot 2 + 5 \cdot 4 = 10 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{30} \]
    |u|.Reviewed
  3. \[ 6 \]
    |v|.Reviewed
  4. \[ \frac{\sqrt{30}}{18} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 10,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{30}}{18} \right)} \approx 72.28^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.51

easy
Let \( \displaystyle \mathbf{u} = \langle 4, 1, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, -4, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 1 \left(-4\right) - 5 \left(-3\right) + 4 \cdot 5 = 31 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{42} \]
    |u|.Reviewed
  3. \[ 5 \sqrt{2} \]
    |v|.Reviewed
  4. \[ \frac{31 \sqrt{21}}{210} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 31,\quad \theta = \operatorname{acos}{\left(\frac{31 \sqrt{21}}{210} \right)} \approx 47.43^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.52

easy
Let \( \displaystyle \mathbf{u} = \langle -4, 0, -2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -3, -5, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 0 \left(-5\right) - 2 \left(-5\right) - 4 \left(-3\right) = 22 \]
    Multiply componentwise and add.✓ Proved
  2. \[ 2 \sqrt{5} \]
    |u|.Reviewed
  3. \[ \sqrt{59} \]
    |v|.Reviewed
  4. \[ \frac{11 \sqrt{295}}{295} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 22,\quad \theta = \operatorname{acos}{\left(\frac{11 \sqrt{295}}{295} \right)} \approx 50.18^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.56

easy
Let \( \displaystyle \mathbf{u} = \langle 0, -4, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, -1, -4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 0 \cdot 5 - 4 \left(-1\right) - 4 \left(-4\right) = 20 \]
    Multiply componentwise and add.✓ Proved
  2. \[ 4 \sqrt{2} \]
    |u|.Reviewed
  3. \[ \sqrt{42} \]
    |v|.Reviewed
  4. \[ \frac{5 \sqrt{21}}{42} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 20,\quad \theta = \operatorname{acos}{\left(\frac{5 \sqrt{21}}{42} \right)} \approx 56.94^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.60

easy
Let \( \displaystyle \mathbf{u} = \langle 0, 1, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 1, 3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-5\right) 3 + 0 \left(-1\right) + 1 \cdot 1 = -14 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{26} \]
    |u|.Reviewed
  3. \[ \sqrt{11} \]
    |v|.Reviewed
  4. \[ - \frac{7 \sqrt{286}}{143} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -14,\quad \theta = \operatorname{acos}{\left(- \frac{7 \sqrt{286}}{143} \right)} \approx 145.88^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.92

easy
Let \( \displaystyle \mathbf{u} = \langle 0, 3, -2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 0, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-2\right) 2 + 0 \cdot 5 + 3 \cdot 0 = -4 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{13} \]
    |u|.Reviewed
  3. \[ \sqrt{29} \]
    |v|.Reviewed
  4. \[ - \frac{4 \sqrt{377}}{377} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -4,\quad \theta = \operatorname{acos}{\left(- \frac{4 \sqrt{377}}{377} \right)} \approx 101.89^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.93

easy
Let \( \displaystyle \mathbf{u} = \langle -5, 0, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -1, 0 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-5\right) 2 + 0 \left(-1\right) - 4 \cdot 0 = -10 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{41} \]
    |u|.Reviewed
  3. \[ \sqrt{5} \]
    |v|.Reviewed
  4. \[ - \frac{2 \sqrt{205}}{41} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -10,\quad \theta = \operatorname{acos}{\left(- \frac{2 \sqrt{205}}{41} \right)} \approx 134.30^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.118

easy
Let \( \displaystyle \mathbf{u} = \langle 5, 0, 4 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, -5, 1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ 5 \left(-4\right) + 0 \left(-5\right) + 4 \cdot 1 = -16 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{41} \]
    |u|.Reviewed
  3. \[ \sqrt{42} \]
    |v|.Reviewed
  4. \[ - \frac{8 \sqrt{1722}}{861} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -16,\quad \theta = \operatorname{acos}{\left(- \frac{8 \sqrt{1722}}{861} \right)} \approx 112.68^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.120

easy
Let \( \displaystyle \mathbf{u} = \langle -5, -2, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle -3, 5, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-2\right) 5 + 1 \cdot 2 - 5 \left(-3\right) = 7 \]
    Multiply componentwise and add.✓ Proved
  2. \[ \sqrt{30} \]
    |u|.Reviewed
  3. \[ \sqrt{38} \]
    |v|.Reviewed
  4. \[ \frac{7 \sqrt{285}}{570} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 7,\quad \theta = \operatorname{acos}{\left(\frac{7 \sqrt{285}}{570} \right)} \approx 78.03^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 9.122

easy
Let \( \displaystyle \mathbf{u} = \langle 1, -5, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, 4, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
  1. \[ \left(-5\right) 4 - 1 \cdot 5 + 1 \cdot 3 = -22 \]
    Multiply componentwise and add.✓ Proved
  2. \[ 3 \sqrt{3} \]
    |u|.Reviewed
  3. \[ 5 \sqrt{2} \]
    |v|.Reviewed
  4. \[ - \frac{11 \sqrt{6}}{45} \]
    cos θ = u·v / (|u||v|).Reviewed
  5. θ = arccos of that.
    Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -22,\quad \theta = \operatorname{acos}{\left(- \frac{11 \sqrt{6}}{45} \right)} \approx 126.78^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More dot product and angles practice

Let \( \displaystyle \mathbf{u} = \langle -3, 4, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, -2, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.5easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -2, 0, -2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, 4, -2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.8easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 2, 1, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 3, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.9easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 0, -1, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -2, 1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.10easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -2, 1, -2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 1, 5, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.11easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -1, -3, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, -1, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.12easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 4, -4, 4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -5, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.13easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -1, 2, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, -5, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.49easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 1, 4, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, 0, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.50easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -3, 5, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 1, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.53easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -4, 5, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, 1, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.54easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 0, 5, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, -1, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.55easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle -4, 3, 4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, -2, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.57easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle -4, 2, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 4, -2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.58easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -1, 0, 3 \rangle \) and \( \displaystyle \mathbf{v} = \langle -3, -5, 3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.59easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 1, -2, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 2, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.91easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 1, 5, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, 4, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.94easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 0, -2, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, -4, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.95easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -1, 5, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle -5, 4, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.96easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 0, -2, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, 5, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.97easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 2, -3, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, 0, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.98easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 5, -5, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, 4, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.115easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -2, 1, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle -3, 1, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.116easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -1, 0, 3 \rangle \) and \( \displaystyle \mathbf{v} = \langle -3, -4, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.117easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 1, -3, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, -1, 0 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.119easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -5, -2, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, -1, -4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.121easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -2, 4, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 3, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.139easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -3, 1, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, -5, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.140easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 2, -2, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, 2, 0 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.141easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 3, 0, -2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, -5, -2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.142easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 2, -2, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle -2, -3, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.143easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 3, 5, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, 5, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.144easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 0, 1, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, -3, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.145easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 4, 0, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, 4, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.146easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 4, 5, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, 4, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.163easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 3, -4, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, -2, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.164easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle -1, 1, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -3, 4, 1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.165easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 4, 3, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, 4, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.166easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -1, -3, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 1, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.167easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 0, 2, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, 1, -2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.168easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 2, 4, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, 2, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.169easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 1, -5, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 1, 3, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.170easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 5, -4, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, -3, -3 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.187easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle -2, -4, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 1, 0 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.188easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 2, -1, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, -4, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.189easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle 4, -2, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle -5, 0, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.190easy✓ Every equation proved
Let \( \displaystyle \mathbf{u} = \langle -3, 1, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, -5, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.191easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 4, -3, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, -1, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.192easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 0, 5, 1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -5, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.193easy✓ Nihil obstat
Let \( \displaystyle \mathbf{u} = \langle 5, -5, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 1, -1, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
Problem 9.194easy✓ Nihil obstat