Question
PreCalculus B - Unit 5 Lesson 6 Portfolio (20 points)
Instructions: Complete the following tasks.
Submission Requirements
Complete all calculations and clearly explain your reasoning.
Use diagrams where necessary to illustrate your answers.
Provide real-world interpretations of your answers, particularly for TASK 3 and Reflections.
TASK 1: Finding the Angle Between Two Vectors (6 pts)
Given the vectors:
a=<3,4>, b=<6,8>
Compute the dot product ab.
Find the magnitude of each vector |a| and |b|.
Use the formula to determine the angle between the vectors.
cos = a b||a||||b||
NOTE: Numerator is DOT PRODUCT. Denominator is MULTIPLICATION
Interpret the result: What does this angle tell us about the relationship between the vectors?
TASK 2: Orthogonal, Parallel, or Neither? (6 pts)
For each of the following pairs of vectors, determine the angle between them and explain if they are orthogonal, parallel vectors or neither.
c=<2,-1> and d=<-4,2>
e=<5,10> and f=<1,2>
g=<7,-3> and h=<-3,-7>
TASK 3: Application - Navigation (6 pts)
A boat is moving along the vector v=<4,3> and the current in the water is represented by the vector
w=<-2,1>.
Find the dot product of the two vectors.
Determine the angle between the boat’s path and the water current.
What does this angle tell us about the effect of the current on the boat?
Suppose the boat must travel in the direction of v, but it needs to counteract the effect of w.
What adjustment should be made?
Reflection: (2 pts)
Answer one of the following in paragraph form. (3-5 sentences)
How might these concepts apply to careers in physics, engineering, or aviation?
In what ways do we encounter vector relationships in daily life, even without realizing it?
How does understanding angles between forces help in problem-solving across different disciplines?
Instructions: Complete the following tasks.
Submission Requirements
Complete all calculations and clearly explain your reasoning.
Use diagrams where necessary to illustrate your answers.
Provide real-world interpretations of your answers, particularly for TASK 3 and Reflections.
TASK 1: Finding the Angle Between Two Vectors (6 pts)
Given the vectors:
a=<3,4>, b=<6,8>
Compute the dot product ab.
Find the magnitude of each vector |a| and |b|.
Use the formula to determine the angle between the vectors.
cos = a b||a||||b||
NOTE: Numerator is DOT PRODUCT. Denominator is MULTIPLICATION
Interpret the result: What does this angle tell us about the relationship between the vectors?
TASK 2: Orthogonal, Parallel, or Neither? (6 pts)
For each of the following pairs of vectors, determine the angle between them and explain if they are orthogonal, parallel vectors or neither.
c=<2,-1> and d=<-4,2>
e=<5,10> and f=<1,2>
g=<7,-3> and h=<-3,-7>
TASK 3: Application - Navigation (6 pts)
A boat is moving along the vector v=<4,3> and the current in the water is represented by the vector
w=<-2,1>.
Find the dot product of the two vectors.
Determine the angle between the boat’s path and the water current.
What does this angle tell us about the effect of the current on the boat?
Suppose the boat must travel in the direction of v, but it needs to counteract the effect of w.
What adjustment should be made?
Reflection: (2 pts)
Answer one of the following in paragraph form. (3-5 sentences)
How might these concepts apply to careers in physics, engineering, or aviation?
In what ways do we encounter vector relationships in daily life, even without realizing it?
How does understanding angles between forces help in problem-solving across different disciplines?
Answers
There are no human answers yet.
Answered by
GPT-4o mini
### TASK 1: Finding the Angle Between Two Vectors (6 pts)
#### Given Vectors:
- \( a = \langle 3, 4 \rangle \)
- \( b = \langle 6, 8 \rangle \)
#### Step 1: Compute the dot product \( a \cdot b \)
\[
a \cdot b = 3 \cdot 6 + 4 \cdot 8 = 18 + 32 = 50
\]
#### Step 2: Find the magnitudes of each vector \( |a| \) and \( |b| \)
\[
|a| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
\[
|b| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
#### Step 3: Use the formula to determine the angle \( \theta \) between the vectors
\[
\cos(\theta) = \frac{a \cdot b}{|a| |b|} = \frac{50}{5 \cdot 10} = \frac{50}{50} = 1
\]
Thus,
\[
\theta = \cos^{-1}(1) = 0^\circ
\]
#### Interpretation:
The angle between the vectors \( a \) and \( b \) is \( 0^\circ \). This tells us that the vectors are pointing in the same direction, indicating they are parallel and that \( b \) is a scalar multiple of \( a \) (specifically, \( b = 2a \)).
---
### TASK 2: Orthogonal, Parallel, or Neither? (6 pts)
1. **For vectors \( c = \langle 2, -1 \rangle \) and \( d = \langle -4, 2 \rangle \)**:
- Dot product:
\[
c \cdot d = 2 \cdot (-4) + (-1) \cdot 2 = -8 - 2 = -10
\]
- Magnitudes:
\[
|c| = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
\[
|d| = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20}
\]
- Thus, they are neither orthogonal (since \( c \cdot d \neq 0 \)) nor parallel (not scalar multiples).
2. **For vectors \( e = \langle 5, 10 \rangle \) and \( f = \langle 1, 2 \rangle \)**:
- Dot product:
\[
e \cdot f = 5 \cdot 1 + 10 \cdot 2 = 5 + 20 = 25
\]
- Magnitudes:
\[
|e| = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125}
\]
\[
|f| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- They are parallel since \( f = \frac{1}{5} e \).
3. **For vectors \( g = \langle 7, -3 \rangle \) and \( h = \langle -3, -7 \rangle \)**:
- Dot product:
\[
g \cdot h = 7 \cdot (-3) + (-3) \cdot (-7) = -21 + 21 = 0
\]
- Thus, they are orthogonal because \( g \cdot h = 0 \).
---
### TASK 3: Application - Navigation (6 pts)
#### Given Vectors:
- The boat's direction: \( v = \langle 4, 3 \rangle \)
- The current's direction: \( w = \langle -2, 1 \rangle \)
#### Step 1: Find the dot product of vectors \( v \) and \( w \)
\[
v \cdot w = 4 \cdot (-2) + 3 \cdot 1 = -8 + 3 = -5
\]
#### Step 2: Calculate the magnitudes
\[
|v| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
\[
|w| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}
\]
#### Step 3: Determine the angle between the vectors
\[
\cos(\theta) = \frac{v \cdot w}{|v| |w|} = \frac{-5}{5 \cdot \sqrt{5}} = -\frac{1}{\sqrt{5}}
\]
Thus,
\[
\theta = \cos^{-1}\left(-\frac{1}{\sqrt{5}}\right) \approx 116.57^\circ
\]
#### Interpretation:
This angle indicates that the boat's path and the water current are directed more than halfway in opposite directions, which means the current may negatively affect the boat's navigation, making it difficult to travel straight in the desired direction.
#### Adjustment to Counteract the Current:
To counteract the current, the boat should adjust its direction slightly towards the positive \( x \) direction to compensate for the current vector’s negative \( x \) component. This may involve sailing at an angle that is adjusted by the current's angle from the north when navigating.
---
### Reflection (2 pts)
#### How does understanding angles between forces help in problem-solving across different disciplines?
Understanding the angles between forces – such as in physics and engineering – is crucial for analyzing the interactions between different vector quantities. For instance, in mechanics, resolving forces into components helps in calculating net forces and respective angles which can affect stability, mobility, and overall efficacy of motion. This principle extends to diverse fields, including aviation, where pilots must account for wind vectors affecting flight paths and navigation, ensuring safe and efficient travel. Moreover, by comprehending these angles, professionals can predict how different forces will affect objects and systems in real-world scenarios, enhancing decision-making processes across multiple applications.
#### Given Vectors:
- \( a = \langle 3, 4 \rangle \)
- \( b = \langle 6, 8 \rangle \)
#### Step 1: Compute the dot product \( a \cdot b \)
\[
a \cdot b = 3 \cdot 6 + 4 \cdot 8 = 18 + 32 = 50
\]
#### Step 2: Find the magnitudes of each vector \( |a| \) and \( |b| \)
\[
|a| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
\[
|b| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
#### Step 3: Use the formula to determine the angle \( \theta \) between the vectors
\[
\cos(\theta) = \frac{a \cdot b}{|a| |b|} = \frac{50}{5 \cdot 10} = \frac{50}{50} = 1
\]
Thus,
\[
\theta = \cos^{-1}(1) = 0^\circ
\]
#### Interpretation:
The angle between the vectors \( a \) and \( b \) is \( 0^\circ \). This tells us that the vectors are pointing in the same direction, indicating they are parallel and that \( b \) is a scalar multiple of \( a \) (specifically, \( b = 2a \)).
---
### TASK 2: Orthogonal, Parallel, or Neither? (6 pts)
1. **For vectors \( c = \langle 2, -1 \rangle \) and \( d = \langle -4, 2 \rangle \)**:
- Dot product:
\[
c \cdot d = 2 \cdot (-4) + (-1) \cdot 2 = -8 - 2 = -10
\]
- Magnitudes:
\[
|c| = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
\[
|d| = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20}
\]
- Thus, they are neither orthogonal (since \( c \cdot d \neq 0 \)) nor parallel (not scalar multiples).
2. **For vectors \( e = \langle 5, 10 \rangle \) and \( f = \langle 1, 2 \rangle \)**:
- Dot product:
\[
e \cdot f = 5 \cdot 1 + 10 \cdot 2 = 5 + 20 = 25
\]
- Magnitudes:
\[
|e| = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125}
\]
\[
|f| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\]
- They are parallel since \( f = \frac{1}{5} e \).
3. **For vectors \( g = \langle 7, -3 \rangle \) and \( h = \langle -3, -7 \rangle \)**:
- Dot product:
\[
g \cdot h = 7 \cdot (-3) + (-3) \cdot (-7) = -21 + 21 = 0
\]
- Thus, they are orthogonal because \( g \cdot h = 0 \).
---
### TASK 3: Application - Navigation (6 pts)
#### Given Vectors:
- The boat's direction: \( v = \langle 4, 3 \rangle \)
- The current's direction: \( w = \langle -2, 1 \rangle \)
#### Step 1: Find the dot product of vectors \( v \) and \( w \)
\[
v \cdot w = 4 \cdot (-2) + 3 \cdot 1 = -8 + 3 = -5
\]
#### Step 2: Calculate the magnitudes
\[
|v| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
\[
|w| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}
\]
#### Step 3: Determine the angle between the vectors
\[
\cos(\theta) = \frac{v \cdot w}{|v| |w|} = \frac{-5}{5 \cdot \sqrt{5}} = -\frac{1}{\sqrt{5}}
\]
Thus,
\[
\theta = \cos^{-1}\left(-\frac{1}{\sqrt{5}}\right) \approx 116.57^\circ
\]
#### Interpretation:
This angle indicates that the boat's path and the water current are directed more than halfway in opposite directions, which means the current may negatively affect the boat's navigation, making it difficult to travel straight in the desired direction.
#### Adjustment to Counteract the Current:
To counteract the current, the boat should adjust its direction slightly towards the positive \( x \) direction to compensate for the current vector’s negative \( x \) component. This may involve sailing at an angle that is adjusted by the current's angle from the north when navigating.
---
### Reflection (2 pts)
#### How does understanding angles between forces help in problem-solving across different disciplines?
Understanding the angles between forces – such as in physics and engineering – is crucial for analyzing the interactions between different vector quantities. For instance, in mechanics, resolving forces into components helps in calculating net forces and respective angles which can affect stability, mobility, and overall efficacy of motion. This principle extends to diverse fields, including aviation, where pilots must account for wind vectors affecting flight paths and navigation, ensuring safe and efficient travel. Moreover, by comprehending these angles, professionals can predict how different forces will affect objects and systems in real-world scenarios, enhancing decision-making processes across multiple applications.
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