Question

In each case, find all \((a, b, c, d)\) in \(\mathbb{R}^{4}\) such that the given set is orthogonal. a. \(\\{(1,2,1,0),(1,-1,1,3),(2,-1,0,-1),\) \((a, b, c, d)\\}\) b. \(\\{(1,0,-1,1),(2,1,1,-1),(1,-3,1,0),\) \((a, b, c, d)\\}\)

Short answer

Solutions involve solving systems of linear equations for each set. Calculate the unknown vector based on orthogonality conditions.

Step-by-step solution

Understanding Orthogonality

Two vectors are orthogonal if their dot product is zero. For a set of vectors to be orthogonal, every pair of vectors in the set must have a dot product of zero.

Orthogonality Check for First Set: Dot Product with (1,2,1,0)

Calculate the dot product between (1,2,1,0) and (a,b,c,d): \(1\times a + 2\times b + 1\times c + 0\times d = a + 2b + c = 0\). This is our first equation: \(a + 2b + c = 0\).

Orthogonality Check for First Set: Dot Product with (1,-1,1,3)

Calculate the dot product between (1,-1,1,3) and (a,b,c,d): \(1\times a + (-1)\times b + 1\times c + 3\times d = a - b + c + 3d = 0\). This gives us our second equation: \(a - b + c + 3d = 0\).

Orthogonality Check for First Set: Dot Product with (2,-1,0,-1)

Calculate the dot product between (2,-1,0,-1) and (a,b,c,d): \(2\times a + (-1)\times b + 0\times c + (-1)\times d = 2a - b - d = 0\). This gives us our third equation: \(2a - b - d = 0\).

Solve Equations for First Set

Using the three equations: 1. \(a + 2b + c = 0\)2. \(a - b + c + 3d = 0\)3. \(2a - b - d = 0\)solve for \(a, b, c, d\). This will require solving a system of linear equations. Substitute and solve the equations step-by-step to find the valid values.

Orthogonality Check for Second Set: Dot Product with (1,0,-1,1)

Calculate the dot product between (1,0,-1,1) and (a,b,c,d): \(1\times a + 0\times b + (-1)\times c + 1\times d = a - c + d = 0\). This is the first equation: \(a - c + d = 0\).

Orthogonality Check for Second Set: Dot Product with (2,1,1,-1)

Calculate the dot product between (2,1,1,-1) and (a,b,c,d): \(2\times a + 1\times b + 1\times c + (-1)\times d = 2a + b + c - d = 0\). This forms our second equation: \(2a + b + c - d = 0\).

Orthogonality Check for Second Set: Dot Product with (1,-3,1,0)

Calculate the dot product between (1,-3,1,0) and (a,b,c,d): \(1\times a + (-3)\times b + 1\times c + 0\times d = a - 3b + c = 0\). This forms our third equation: \(a - 3b + c = 0\).

Solve Equations for Second Set

Using the three equations: 1. \(a - c + d = 0\)2. \(2a + b + c - d = 0\)3. \(a - 3b + c = 0\)solve for \(a, b, c, d\). Use substitution and elimination to find the solution set.

Key concepts

Dot Product

The dot product, also known as the scalar product, is an essential operation in vector spaces. It is a way to multiply two vectors, resulting not in another vector, but a single number (scalar). The formula for the dot product of two vectors \(\mathbf{u} = (u_1, u_2, \ldots, u_n)\) and \(\mathbf{v} = (v_1, v_2, \ldots, v_n)\) is written as: \[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n\]This formula essentially sums up the products of the corresponding components from each vector.
The dot product is particularly useful in determining angles between vectors or understanding the concept of orthogonality. When the dot product of two vectors equals zero, it indicates that the vectors are orthogonal to each other. This property is frequently used to solve problems related to orthogonality in vector spaces, like the exercise given here.
In practice, calculating the dot product is a matter of careful component-wise multiplication and addition.

Solving Linear Equations

When tackling systems of linear equations, such as those needed to solve for variables in a vector that must be orthogonal to given vectors, you are essentially looking to find values that satisfy all equations simultaneously.
Each equation represents a condition that must hold true. For example, if you have equations like: \[a + 2b + c = 0\]and \[a - b + c + 3d = 0\]you need to find values for \(a\), \(b\), \(c\), and \(d\) that simultaneously satisfy all equations.
A common method of solving such systems is substitution or elimination:

• Substitution involves isolating one variable in one equation and substituting this expression into another equation.
• Elimination involves adding or subtracting equations from each other to eliminate one variable, reducing the system to fewer variables.

This process continues until solutions for all variables are found.
In vector spaces, solvability ensures the orthogonality condition is met for all required vector pairs.

Vector Spaces

Vector spaces, or linear spaces, are a core concept in linear algebra. They provide a framework for understanding vectors, vector addition, and scalar multiplication. A vector space consists of vectors, a set of operations (usually addition and scalar multiplication), and adheres to certain axioms such as associativity, commutativity, and distributivity.
In general, any object that follows these rules can be part of a vector space, making the concept incredibly versatile across mathematics and its applications. Some key elements of vector spaces include:

• Closure under addition: The sum of any two vectors in the space results in another vector in the same space.
• Closure under scalar multiplication: Multiplying any vector in the space by a scalar results in another vector in the space.
• Existence of zero vector: There is a vector, often denoted as \(\mathbf{0}\), which acts as an additive identity.

These characteristics make vector spaces a foundation for exploring solutions in linear algebra, including when determining sets of orthogonal vectors.

Orthogonal Vectors

Orthogonal vectors are fundamental in understanding angles and dimensions within vector spaces. Two vectors are orthogonal if their dot product is zero, which geometrically means they form a right angle ( ext{90 degrees}) with each other.
Orthogonality can be extended to sets of vectors, where all possible pairs are orthogonal to each other. Such sets are important in decomposing spaces into simpler, more manageable components, often referenced in topics like Fourier analysis and orthogonal bases. Features of orthogonal vectors:

• Independent Orientation: In a Euclidean space, orthogonal vectors imply that the directions are independent of each other.
• Orthogonal Sets: A collection of vectors where every pair is orthogonal forms what is known as an orthogonal set.
• Zero Dot Product: As mentioned earlier, the defining mathematical condition is \(\mathbf{u} \cdot \mathbf{v} = 0\) for vectors \(\mathbf{u}\) and \(\mathbf{v}\).

The principle of orthogonality is central to many applications in physics and engineering, where systems often exploit these independent dimensions of movement or force without interference.