**Title:** Orthogonality of Vectors **Question:** Determine which set of vectors is orthogonal. **Options:** 1. **Option A:** - **Vector V:** ⟨3, 1⟩ - **Vector W:** ⟨2, -6⟩ 2. **Option B:** - **Vector V:** ⟨-10, 5⟩ - **Vector W:** ⟨1, -2⟩ 3. **Option C:** - **Vector V:** ⟨10, 2⟩ - **Vector W:** ⟨-5, -1⟩ 4. **Option D:** - **Vector V:** ⟨3, -1⟩ - **Vector W:** ⟨2, -6⟩ **Explanation:** Orthogonality of vectors is determined by their dot product. Two vectors **V** and **W** are orthogonal if and only if their dot product is zero (\( V \cdot W = 0 \)). ### How to Determine Vector Orthogonality: - The dot product of two vectors \( V = \langle a_1, a_2 \rangle \) and \( W = \langle b_1, b_2 \rangle \) is given by: \[ V \cdot W = a_1 \cdot b_1 + a_2 \cdot b_2 \] - Calculate the dot product for each option to determine the orthogonality. ### Example Calculation: **For Option A:** \( V = \langle 3, 1 \rangle \) \( W = \langle 2, -6 \rangle \) \[ V \cdot W = (3 \times 2) + (1 \times -6) = 6 - 6 = 0 \] Since the dot product is zero, vectors in Option A are orthogonal.
**Title:** Orthogonality of Vectors **Question:** Determine which set of vectors is orthogonal. **Options:** 1. **Option A:** - **Vector V:** ⟨3, 1⟩ - **Vector W:** ⟨2, -6⟩ 2. **Option B:** - **Vector V:** ⟨-10, 5⟩ - **Vector W:** ⟨1, -2⟩ 3. **Option C:** - **Vector V:** ⟨10, 2⟩ - **Vector W:** ⟨-5, -1⟩ 4. **Option D:** - **Vector V:** ⟨3, -1⟩ - **Vector W:** ⟨2, -6⟩ **Explanation:** Orthogonality of vectors is determined by their dot product. Two vectors **V** and **W** are orthogonal if and only if their dot product is zero (\( V \cdot W = 0 \)). ### How to Determine Vector Orthogonality: - The dot product of two vectors \( V = \langle a_1, a_2 \rangle \) and \( W = \langle b_1, b_2 \rangle \) is given by: \[ V \cdot W = a_1 \cdot b_1 + a_2 \cdot b_2 \] - Calculate the dot product for each option to determine the orthogonality. ### Example Calculation: **For Option A:** \( V = \langle 3, 1 \rangle \) \( W = \langle 2, -6 \rangle \) \[ V \cdot W = (3 \times 2) + (1 \times -6) = 6 - 6 = 0 \] Since the dot product is zero, vectors in Option A are orthogonal.
Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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Question
![**Title:** Orthogonality of Vectors
**Question:** Determine which set of vectors is orthogonal.
**Options:**
1. **Option A:**
- **Vector V:** ⟨3, 1⟩
- **Vector W:** ⟨2, -6⟩
2. **Option B:**
- **Vector V:** ⟨-10, 5⟩
- **Vector W:** ⟨1, -2⟩
3. **Option C:**
- **Vector V:** ⟨10, 2⟩
- **Vector W:** ⟨-5, -1⟩
4. **Option D:**
- **Vector V:** ⟨3, -1⟩
- **Vector W:** ⟨2, -6⟩
**Explanation:** Orthogonality of vectors is determined by their dot product. Two vectors **V** and **W** are orthogonal if and only if their dot product is zero (\( V \cdot W = 0 \)).
### How to Determine Vector Orthogonality:
- The dot product of two vectors \( V = \langle a_1, a_2 \rangle \) and \( W = \langle b_1, b_2 \rangle \) is given by:
\[
V \cdot W = a_1 \cdot b_1 + a_2 \cdot b_2
\]
- Calculate the dot product for each option to determine the orthogonality.
### Example Calculation:
**For Option A:**
\( V = \langle 3, 1 \rangle \)
\( W = \langle 2, -6 \rangle \)
\[
V \cdot W = (3 \times 2) + (1 \times -6) = 6 - 6 = 0
\]
Since the dot product is zero, vectors in Option A are orthogonal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb65d250f-ce79-4155-be29-8b1588e7e3e2%2F07ecc717-58a4-4bc6-a402-bfd82321fd05%2Fnwwzheph_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Orthogonality of Vectors
**Question:** Determine which set of vectors is orthogonal.
**Options:**
1. **Option A:**
- **Vector V:** ⟨3, 1⟩
- **Vector W:** ⟨2, -6⟩
2. **Option B:**
- **Vector V:** ⟨-10, 5⟩
- **Vector W:** ⟨1, -2⟩
3. **Option C:**
- **Vector V:** ⟨10, 2⟩
- **Vector W:** ⟨-5, -1⟩
4. **Option D:**
- **Vector V:** ⟨3, -1⟩
- **Vector W:** ⟨2, -6⟩
**Explanation:** Orthogonality of vectors is determined by their dot product. Two vectors **V** and **W** are orthogonal if and only if their dot product is zero (\( V \cdot W = 0 \)).
### How to Determine Vector Orthogonality:
- The dot product of two vectors \( V = \langle a_1, a_2 \rangle \) and \( W = \langle b_1, b_2 \rangle \) is given by:
\[
V \cdot W = a_1 \cdot b_1 + a_2 \cdot b_2
\]
- Calculate the dot product for each option to determine the orthogonality.
### Example Calculation:
**For Option A:**
\( V = \langle 3, 1 \rangle \)
\( W = \langle 2, -6 \rangle \)
\[
V \cdot W = (3 \times 2) + (1 \times -6) = 6 - 6 = 0
\]
Since the dot product is zero, vectors in Option A are orthogonal.
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