Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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![### Problem 12: Finding the Angle Between Vectors
Given the following conditions:
- The dot product of vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{a} \cdot \vec{b} = \sqrt{3}\),
- The cross product of vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{a} \times \vec{b} = \langle 1, 2, 2 \rangle\),
Find the angle between vectors \(\vec{a}\) and \(\vec{b}\).
#### Solution:
- Start by understanding the dot product, which is given by:
\[
\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta
\]
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\).
- The magnitude of the cross product \(\vec{a}\times\vec{b}\) is given by:
\[
|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta
\]
Here, \(|\vec{a} \times \vec{b}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3\).
- Use the relationships between the dot product and cross product to find the angle \(\theta\).
This problem illustrates fundamental vector operations crucial for understanding physics and engineering concepts such as force, momentum, and more.](https://content.bartleby.com/qna-images/question/18107b7c-8d8e-412b-a2cd-6addf4186be2/7efe6fdd-d4b6-4ed9-a517-e23e79247ebb/rzp5lc_processed.png)
Transcribed Image Text:### Problem 12: Finding the Angle Between Vectors
Given the following conditions:
- The dot product of vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{a} \cdot \vec{b} = \sqrt{3}\),
- The cross product of vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{a} \times \vec{b} = \langle 1, 2, 2 \rangle\),
Find the angle between vectors \(\vec{a}\) and \(\vec{b}\).
#### Solution:
- Start by understanding the dot product, which is given by:
\[
\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta
\]
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\).
- The magnitude of the cross product \(\vec{a}\times\vec{b}\) is given by:
\[
|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta
\]
Here, \(|\vec{a} \times \vec{b}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3\).
- Use the relationships between the dot product and cross product to find the angle \(\theta\).
This problem illustrates fundamental vector operations crucial for understanding physics and engineering concepts such as force, momentum, and more.
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