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To find the angle between the vectors [tex]\(\underset{\sim}{a} = 2 \underset{\sim}{i} + \underset{\sim}{j} + \underset{\sim}{k}\)[/tex] and [tex]\(\underset{\sim}{b} = \underset{\sim}{i} - \underset{\sim}{j} + 3 \underset{\sim}{k}\)[/tex], we need to follow several steps:
1. Write down the vectors:
- [tex]\(\underset{\sim}{a} = 2 \underset{\sim}{i} + \underset{\sim}{j} + \underset{\sim}{k}\)[/tex]
- [tex]\(\underset{\sim}{b} = \underset{\sim}{i} - \underset{\sim}{j} + 3 \underset{\sim}{k}\)[/tex]
2. Compute the dot product of the vectors:
The dot product of two vectors [tex]\(\underset{\sim}{a} = [a_1, a_2, a_3]\)[/tex] and [tex]\(\underset{\sim}{b} = [b_1, b_2, b_3]\)[/tex] is given by:
[tex]\[ \underset{\sim}{a} \cdot \underset{\sim}{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \][/tex]
Here,
[tex]\[ \underset{\sim}{a} = [2, 1, 1] \][/tex]
[tex]\[ \underset{\sim}{b} = [1, -1, 3] \][/tex]
The dot product is:
[tex]\[ 2 \cdot 1 + 1 \cdot (-1) + 1 \cdot 3 = 2 - 1 + 3 = 4 \][/tex]
3. Compute the magnitudes of the vectors:
The magnitude of a vector [tex]\(\underset{\sim}{a} = [a_1, a_2, a_3]\)[/tex] is given by:
[tex]\[ ||\underset{\sim}{a}|| = \sqrt{a_1^2 + a_2^2 + a_3^2} \][/tex]
For [tex]\(\underset{\sim}{a}\)[/tex]:
[tex]\[ ||\underset{\sim}{a}|| = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \approx 2.449 \][/tex]
For [tex]\(\underset{\sim}{b}\)[/tex]:
[tex]\[ ||\underset{\sim}{b}|| = \sqrt{1^2 + (-1)^2 + 3^2} = \sqrt{1 + 1 + 9} = \sqrt{11} \approx 3.317 \][/tex]
4. Calculate the cosine of the angle between the vectors:
The cosine of the angle [tex]\(\theta\)[/tex] between the vectors is given by:
[tex]\[ \cos \theta = \frac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{||\underset{\sim}{a}|| \, ||\underset{\sim}{b}||} \][/tex]
Substituting the values we found:
[tex]\[ \cos \theta = \frac{4}{2.449 \cdot 3.317} \approx \frac{4}{8} = 0.492 \][/tex]
5. Compute the angle in radians and degrees:
The angle [tex]\(\theta\)[/tex] can be found by taking the arccosine of the cosine value:
[tex]\[ \theta = \arccos(0.492) \approx 1.056 \text{ radians} \][/tex]
To convert the angle from radians to degrees:
[tex]\[ \theta_{degrees} = \theta_{radians} \times \left(\frac{180}{\pi}\right) \approx 1.056 \times 57.2958 \approx 60.504^\circ \][/tex]
Summary:
- Dot product: [tex]\(4\)[/tex]
- Magnitude of [tex]\(\underset{\sim}{a}\)[/tex]: [tex]\(2.449\)[/tex]
- Magnitude of [tex]\(\underset{\sim}{b}\)[/tex]: [tex]\(3.317\)[/tex]
- Cosine of the angle: [tex]\(0.492\)[/tex]
- Angle in radians: [tex]\(1.056\)[/tex]
- Angle in degrees: [tex]\(60.504^\circ\)[/tex]
Therefore, the angle between the vectors [tex]\(\underset{\sim}{a}\)[/tex] and [tex]\(\underset{\sim}{b}\)[/tex] is approximately [tex]\(60.504^\circ\)[/tex].
1. Write down the vectors:
- [tex]\(\underset{\sim}{a} = 2 \underset{\sim}{i} + \underset{\sim}{j} + \underset{\sim}{k}\)[/tex]
- [tex]\(\underset{\sim}{b} = \underset{\sim}{i} - \underset{\sim}{j} + 3 \underset{\sim}{k}\)[/tex]
2. Compute the dot product of the vectors:
The dot product of two vectors [tex]\(\underset{\sim}{a} = [a_1, a_2, a_3]\)[/tex] and [tex]\(\underset{\sim}{b} = [b_1, b_2, b_3]\)[/tex] is given by:
[tex]\[ \underset{\sim}{a} \cdot \underset{\sim}{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \][/tex]
Here,
[tex]\[ \underset{\sim}{a} = [2, 1, 1] \][/tex]
[tex]\[ \underset{\sim}{b} = [1, -1, 3] \][/tex]
The dot product is:
[tex]\[ 2 \cdot 1 + 1 \cdot (-1) + 1 \cdot 3 = 2 - 1 + 3 = 4 \][/tex]
3. Compute the magnitudes of the vectors:
The magnitude of a vector [tex]\(\underset{\sim}{a} = [a_1, a_2, a_3]\)[/tex] is given by:
[tex]\[ ||\underset{\sim}{a}|| = \sqrt{a_1^2 + a_2^2 + a_3^2} \][/tex]
For [tex]\(\underset{\sim}{a}\)[/tex]:
[tex]\[ ||\underset{\sim}{a}|| = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \approx 2.449 \][/tex]
For [tex]\(\underset{\sim}{b}\)[/tex]:
[tex]\[ ||\underset{\sim}{b}|| = \sqrt{1^2 + (-1)^2 + 3^2} = \sqrt{1 + 1 + 9} = \sqrt{11} \approx 3.317 \][/tex]
4. Calculate the cosine of the angle between the vectors:
The cosine of the angle [tex]\(\theta\)[/tex] between the vectors is given by:
[tex]\[ \cos \theta = \frac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{||\underset{\sim}{a}|| \, ||\underset{\sim}{b}||} \][/tex]
Substituting the values we found:
[tex]\[ \cos \theta = \frac{4}{2.449 \cdot 3.317} \approx \frac{4}{8} = 0.492 \][/tex]
5. Compute the angle in radians and degrees:
The angle [tex]\(\theta\)[/tex] can be found by taking the arccosine of the cosine value:
[tex]\[ \theta = \arccos(0.492) \approx 1.056 \text{ radians} \][/tex]
To convert the angle from radians to degrees:
[tex]\[ \theta_{degrees} = \theta_{radians} \times \left(\frac{180}{\pi}\right) \approx 1.056 \times 57.2958 \approx 60.504^\circ \][/tex]
Summary:
- Dot product: [tex]\(4\)[/tex]
- Magnitude of [tex]\(\underset{\sim}{a}\)[/tex]: [tex]\(2.449\)[/tex]
- Magnitude of [tex]\(\underset{\sim}{b}\)[/tex]: [tex]\(3.317\)[/tex]
- Cosine of the angle: [tex]\(0.492\)[/tex]
- Angle in radians: [tex]\(1.056\)[/tex]
- Angle in degrees: [tex]\(60.504^\circ\)[/tex]
Therefore, the angle between the vectors [tex]\(\underset{\sim}{a}\)[/tex] and [tex]\(\underset{\sim}{b}\)[/tex] is approximately [tex]\(60.504^\circ\)[/tex].
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