Find All (a, B, C, D) In R4 Such That The Given Set Is Orthogonal. {(1, 2, 1, 0), (1, 1, 1, 3), (2, 1,
Introduction to Orthogonality in R4
Understanding the concept of orthogonality in vector spaces is crucial in fields such as linear algebra, engineering, computer science, and mathematics. When vectors are orthogonal, they are perpendicular in a geometric sense, meaning their dot product equals zero. This property is fundamental in applications like orthogonal projections, signal processing, and data analysis.
In this article, we focus on a specific problem: finding all possible values of (a, B, C, D) in R4 such that a given set of vectors is orthogonal. The set includes vectors with some unknown components, and our goal is to determine all these components that satisfy the orthogonality conditions.
Understanding the Problem Statement
The problem involves a set of vectors in four-dimensional space, R4:
- Vector 1: (1, 2, 1, 0)
- Vector 2: (1, 1, 1, 3)
- Vector 3: (2, 1, a, B)
- Vector 4: (C, D, ..., ...)
(Note: The original problem seems to be incomplete, but we proceed with the assumption that the third vector involves parameters a and B, and the fourth involves C and D.)
Our goal is to find all possible values of a, B, C, and D such that these vectors are mutually orthogonal.
Key concepts:
- Orthogonality: Two vectors u and v are orthogonal if their dot product u · v = 0.
- Mutual Orthogonality: All pairs of vectors in the set are orthogonal to each other.
Given that, the conditions for orthogonality among the vectors are:
- u₁ · u₂ = 0
- u₁ · u₃ = 0
- u₁ · u₄ = 0
- u₂ · u₃ = 0
- u₂ · u₄ = 0
- u₃ · u₄ = 0
By solving these equations, we can find all the possible values of (a, B, C, D).
Step-by-Step Solution Approach
To solve for the unknown components, follow these steps:
- Write down the dot product equations for each pair of vectors.
- Substitute the known vectors and parameters.
- Solve the resulting system of equations for the unknowns.
- Determine all solutions that satisfy the orthogonality conditions.
Let's proceed with the detailed calculations.
Step 1: Dot Product of Known Vectors
Vectors:
- u₁ = (1, 2, 1, 0)
- u₂ = (1, 1, 1, 3)
- u₃ = (2, 1, a, B)
- u₄ = (C, D, ..., ...)
(Assuming u₄ is a vector with unknown components C and D.)
Note: Since the problem statement is incomplete, for the sake of completeness, we'll assume the fourth vector is (C, D, E, F), where E and F are additional unknowns. But as per the initial problem, only a, B, C, D are variables, so perhaps u₄ is (C, D, 0, 0). For simplicity, we assume u₄ = (C, D, 0, 0).
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Dot product equations:
- u₁ · u₂ = (1)(1) + (2)(1) + (1)(1) + (0)(3) = 1 + 2 + 1 + 0 = 4
Since this is not zero, the vectors u₁ and u₂ are not orthogonal unless we interpret the problem differently. But the goal is to find the unknowns such that the entire set is orthogonal, which involves the other pairs.
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Assumption: The initial set may be intended to be mutually orthogonal, and perhaps the first two vectors are fixed, while the third and fourth vectors are parameterized.
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Step 2: Define the Conditions for Orthogonality
Assuming the vectors are:
- u₁ = (1, 2, 1, 0)
- u₂ = (1, 1, 1, 3)
- u₃ = (2, 1, a, B)
- u₄ = (C, D, 0, 0)
The orthogonality conditions are:
- u₁ · u₂ = 0
- u₁ · u₃ = 0
- u₁ · u₄ = 0
- u₂ · u₃ = 0
- u₂ · u₄ = 0
- u₃ · u₄ = 0
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Step 3: Write Out Each Dot Product Equation
- u₁ · u₂ = 0
(1)(1) + (2)(1) + (1)(1) + (0)(3) = 1 + 2 + 1 + 0 = 4
Result: 4 ≠ 0, so vectors u₁ and u₂ are not orthogonal unless the problem expects the set to be orthogonal excluding the pair (u₁, u₂), or perhaps the initial vectors are different.
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Clarification:
Given the initial confusion, perhaps the problem is to find (a, B, C, D) such that the vectors:
- v₁ = (1, 2, 1, 0)
- v₂ = (1, 1, 1, 3)
- v₃ = (2, 1, a, B)
- v₄ = (C, D, 0, 0)
are mutually orthogonal except for the pair (v₁, v₂). Alternatively, perhaps the problem wants the set to be orthogonal with respect to these vectors, and the initial set is just a starting point.
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Step 4: Focus on the core problem — solving for (a, B, C, D)
Assuming the goal is to find (a, B, C, D) such that the following conditions hold:
- u₁ · u₃ = 0
- u₁ · u₄ = 0
- u₂ · u₃ = 0
- u₂ · u₄ = 0
- u₃ · u₄ = 0
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Step 5: Set Up Equations and Solve
Equation 1: u₁ · u₃ = 0
(1)(2) + (2)(1) + (1)(a) + (0)(B) = 2 + 2 + a + 0 = 4 + a = 0
=> a = -4
Equation 2: u₁ · u₄ = 0
(1)(C) + (2)(D) + (1)(0) + (0)(0) = C + 2D + 0 + 0 = C + 2D = 0
=> C = -2D
Equation 3: u₂ · u₃ = 0
(1)(2) + (1)(1) + (1)(a) + (3)(B) = 2 + 1 + a + 3B
Substitute a = -4:
2 + 1 - 4 + 3B = (-1) + 3B
Set equal to zero:
-1 + 3B = 0
=> 3B = 1
=> B = 1/3
Equation 4: u₂ · u₄ = 0
(1)(C) + (1)(D) + (1)(0) + (3)(0) = C + D + 0 + 0 = C + D
Recall C = -2D from earlier:
C + D = -2D + D = -D
Set equal to zero:
-D = 0
=> D = 0
Then C = -2D = 0
Equation 5: u₃ · u₄ = 0
(2)(C) + (1)(D) + (a)(0) + (B)(0) = 2C + D + 0 + 0 = 2C + D
Recall C = 0 and D = 0:
20 + 0 = 0
Equation satisfied.
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Final Solution:
| Variable | Value |
|------------|---------|
| a | -4 |
| B | 1/3 |
| C | 0 |
| D | 0 |
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Conclusion and Summary
In this detailed analysis, we've demonstrated how to determine the unknown components (a, B, C, D) in R4 such that the set of vectors is mutually
