Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Determine which set of vectors is orthogonal.
- **Option A:** \( \mathbf{v} = (0, -2), \mathbf{w} = \left( 0, \frac{3}{2} \right) \)
- **Option B:** \( \mathbf{v} = \left( -\frac{3}{2}, -2 \right), \mathbf{w} = (1, 5) \)
- **Option C:** \( \mathbf{v} = \left( -\frac{3}{2}, 0 \right), \mathbf{w} = (1, 0) \)
- **Option D:** \( \mathbf{v} = \left( -\frac{3}{2}, 1 \right), \mathbf{w} = \left( 1, \frac{3}{2} \right) \)
### Explanation
To determine which set of vectors is orthogonal, we need to check if the dot product of the two vectors in each set equals zero. Two vectors \(\mathbf{v} = (v_1, v_2)\) and \(\mathbf{w} = (w_1, w_2)\) are orthogonal if:
\[ \mathbf{v} \cdot \mathbf{w} = v_1 \cdot w_1 + v_2 \cdot w_2 = 0 \]
Let's analyze each option:
1. **Option A:**
\[
\mathbf{v} \cdot \mathbf{w} = 0 \cdot 0 + (-2) \cdot \frac{3}{2} = 0 - 3 = -3 \quad (\text{Not orthogonal})
\]
2. **Option B:**
\[
\mathbf{v} \cdot \mathbf{w} = \left( -\frac{3}{2} \right) \cdot 1 + (-2) \cdot 5 = -\frac{3}{2} - 10 = -\frac{23}{2} \quad (\text{Not orthogonal})
\]
3. **Option C:**
\[
\mathbf{v} \cdot \mathbf{w} = \left( -\frac{3}{2} \right](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcccb93a5-bda7-4c84-b750-51d9219067a4%2Fa6609bd2-b43f-4f87-915f-351da94c1dbe%2Fqmf45ge_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determine which set of vectors is orthogonal.
- **Option A:** \( \mathbf{v} = (0, -2), \mathbf{w} = \left( 0, \frac{3}{2} \right) \)
- **Option B:** \( \mathbf{v} = \left( -\frac{3}{2}, -2 \right), \mathbf{w} = (1, 5) \)
- **Option C:** \( \mathbf{v} = \left( -\frac{3}{2}, 0 \right), \mathbf{w} = (1, 0) \)
- **Option D:** \( \mathbf{v} = \left( -\frac{3}{2}, 1 \right), \mathbf{w} = \left( 1, \frac{3}{2} \right) \)
### Explanation
To determine which set of vectors is orthogonal, we need to check if the dot product of the two vectors in each set equals zero. Two vectors \(\mathbf{v} = (v_1, v_2)\) and \(\mathbf{w} = (w_1, w_2)\) are orthogonal if:
\[ \mathbf{v} \cdot \mathbf{w} = v_1 \cdot w_1 + v_2 \cdot w_2 = 0 \]
Let's analyze each option:
1. **Option A:**
\[
\mathbf{v} \cdot \mathbf{w} = 0 \cdot 0 + (-2) \cdot \frac{3}{2} = 0 - 3 = -3 \quad (\text{Not orthogonal})
\]
2. **Option B:**
\[
\mathbf{v} \cdot \mathbf{w} = \left( -\frac{3}{2} \right) \cdot 1 + (-2) \cdot 5 = -\frac{3}{2} - 10 = -\frac{23}{2} \quad (\text{Not orthogonal})
\]
3. **Option C:**
\[
\mathbf{v} \cdot \mathbf{w} = \left( -\frac{3}{2} \right
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