College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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4:Need help
![**Problem 4:**
*Given the two vectors, find the dot product and the angle between them.*
\[ \vec{A} = 4\hat{x} - 7\hat{y} \]
\[ \vec{B} = -3\hat{x} + 2\hat{y} - 5\hat{z} \]
In this problem, we need to compute the dot product of vectors \(\vec{A}\) and \(\vec{B}\), and subsequently determine the angle between these two vectors. The vectors \(\vec{A}\) and \(\vec{B}\) are described in terms of their components along the \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\) axes.
### Steps to Solve:
1. **Dot Product Calculation:**
- The dot product formula for two vectors \(\vec{A} = a_1\hat{x} + a_2\hat{y} + a_3\hat{z}\) and \(\vec{B} = b_1\hat{x} + b_2\hat{y} + b_3\hat{z}\) is:
\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \]
- Substitute the given values into the formula.
2. **Angle Between Vectors:**
- Use the formula for the angle \(\theta\) between two vectors:
\[ \cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \]
- Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\):
\[ |\vec{A}| = \sqrt{(a_1)^2 + (a_2)^2 + (a_3)^2} \]
\[ |\vec{B}| = \sqrt{(b_1)^2 + (b_2)^2 + (b_3)^2} \]
- Solve for \(\theta\).
Provide detailed calculations and solutions for each step when using this material on an educational website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F560885b1-95a6-46a1-8479-9132da42c8bd%2Fb855b041-83fe-490e-b8b4-cf5799d187ce%2Fa64qq6f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4:**
*Given the two vectors, find the dot product and the angle between them.*
\[ \vec{A} = 4\hat{x} - 7\hat{y} \]
\[ \vec{B} = -3\hat{x} + 2\hat{y} - 5\hat{z} \]
In this problem, we need to compute the dot product of vectors \(\vec{A}\) and \(\vec{B}\), and subsequently determine the angle between these two vectors. The vectors \(\vec{A}\) and \(\vec{B}\) are described in terms of their components along the \(\hat{x}\), \(\hat{y}\), and \(\hat{z}\) axes.
### Steps to Solve:
1. **Dot Product Calculation:**
- The dot product formula for two vectors \(\vec{A} = a_1\hat{x} + a_2\hat{y} + a_3\hat{z}\) and \(\vec{B} = b_1\hat{x} + b_2\hat{y} + b_3\hat{z}\) is:
\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \]
- Substitute the given values into the formula.
2. **Angle Between Vectors:**
- Use the formula for the angle \(\theta\) between two vectors:
\[ \cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \]
- Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\):
\[ |\vec{A}| = \sqrt{(a_1)^2 + (a_2)^2 + (a_3)^2} \]
\[ |\vec{B}| = \sqrt{(b_1)^2 + (b_2)^2 + (b_3)^2} \]
- Solve for \(\theta\).
Provide detailed calculations and solutions for each step when using this material on an educational website.
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