Question
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>
Dot product: C times d = 2 times (-4) + (-1) times 2 = -8 - 2 = -10
e=<5,10> and f=<1,2>
g=<7,-3> and h=<-3,-7>
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>
Dot product: C times d = 2 times (-4) + (-1) times 2 = -8 - 2 = -10
e=<5,10> and f=<1,2>
g=<7,-3> and h=<-3,-7>
Answers
Answered by
GPT-4o mini
To determine if the pairs of vectors are orthogonal, parallel, or neither, we will calculate the dot product of each pair and analyze the result, as well as check for scalar multiples to determine parallelism.
### Pair 1: \( \mathbf{c} = \langle 2, -1 \rangle \) and \( \mathbf{d} = \langle -4, 2 \rangle \)
1. **Dot Product Calculation:**
\[
\mathbf{c} \cdot \mathbf{d} = (2)(-4) + (-1)(2) = -8 - 2 = -10
\]
2. **Interpretation:**
The dot product is not zero (\(-10\)), so the vectors are **not orthogonal**.
3. **Check for Parallelism:**
To be parallel, one vector must be a scalar multiple of the other.
\[
\mathbf{d} = -2 \cdot \mathbf{c} \quad \text{(since } \langle -4, 2 \rangle = -2 \langle 2, -1 \rangle\text{)}
\]
This means the vectors are indeed **parallel**.
### Pair 2: \( \mathbf{e} = \langle 5, 10 \rangle \) and \( \mathbf{f} = \langle 1, 2 \rangle \)
1. **Dot Product Calculation:**
\[
\mathbf{e} \cdot \mathbf{f} = (5)(1) + (10)(2) = 5 + 20 = 25
\]
2. **Interpretation:**
The dot product is not zero (\(25\)), so the vectors are **not orthogonal**.
3. **Check for Parallelism:**
\(\mathbf{e} = 5 \cdot \mathbf{f} \quad \text{(since } \langle 5, 10 \rangle = 5 \langle 1, 2 \rangle\text{)}\)
This means the vectors are **parallel**.
### Pair 3: \( \mathbf{g} = \langle 7, -3 \rangle \) and \( \mathbf{h} = \langle -3, -7 \rangle \)
1. **Dot Product Calculation:**
\[
\mathbf{g} \cdot \mathbf{h} = (7)(-3) + (-3)(-7) = -21 + 21 = 0
\]
2. **Interpretation:**
The dot product is zero, indicating that the vectors are **orthogonal**.
### Summary:
- Vectors \( \mathbf{c} \) and \( \mathbf{d} \): **Parallel**
- Vectors \( \mathbf{e} \) and \( \mathbf{f} \): **Parallel**
- Vectors \( \mathbf{g} \) and \( \mathbf{h} \): **Orthogonal**
### Pair 1: \( \mathbf{c} = \langle 2, -1 \rangle \) and \( \mathbf{d} = \langle -4, 2 \rangle \)
1. **Dot Product Calculation:**
\[
\mathbf{c} \cdot \mathbf{d} = (2)(-4) + (-1)(2) = -8 - 2 = -10
\]
2. **Interpretation:**
The dot product is not zero (\(-10\)), so the vectors are **not orthogonal**.
3. **Check for Parallelism:**
To be parallel, one vector must be a scalar multiple of the other.
\[
\mathbf{d} = -2 \cdot \mathbf{c} \quad \text{(since } \langle -4, 2 \rangle = -2 \langle 2, -1 \rangle\text{)}
\]
This means the vectors are indeed **parallel**.
### Pair 2: \( \mathbf{e} = \langle 5, 10 \rangle \) and \( \mathbf{f} = \langle 1, 2 \rangle \)
1. **Dot Product Calculation:**
\[
\mathbf{e} \cdot \mathbf{f} = (5)(1) + (10)(2) = 5 + 20 = 25
\]
2. **Interpretation:**
The dot product is not zero (\(25\)), so the vectors are **not orthogonal**.
3. **Check for Parallelism:**
\(\mathbf{e} = 5 \cdot \mathbf{f} \quad \text{(since } \langle 5, 10 \rangle = 5 \langle 1, 2 \rangle\text{)}\)
This means the vectors are **parallel**.
### Pair 3: \( \mathbf{g} = \langle 7, -3 \rangle \) and \( \mathbf{h} = \langle -3, -7 \rangle \)
1. **Dot Product Calculation:**
\[
\mathbf{g} \cdot \mathbf{h} = (7)(-3) + (-3)(-7) = -21 + 21 = 0
\]
2. **Interpretation:**
The dot product is zero, indicating that the vectors are **orthogonal**.
### Summary:
- Vectors \( \mathbf{c} \) and \( \mathbf{d} \): **Parallel**
- Vectors \( \mathbf{e} \) and \( \mathbf{f} \): **Parallel**
- Vectors \( \mathbf{g} \) and \( \mathbf{h} \): **Orthogonal**