Question

Orthogonal vectors Let a and b be real numbers. Find two vectors that are orthogonal to \langle 0,1,1\rangle and to each other.

Short answer

Question: Based on the given vector \langle 0,1,1\rangle, identify two other vectors that are orthogonal to this given vector and also orthogonal to each other. Answer: Two vectors that meet the criteria are u=\langle 1,-1,1 \rangle and v=\langle 1,-\frac{1}{2},-\frac{1}{2} \rangle.

Step-by-step solution

Reviewing Vector Orthogonality

Two vectors are orthogonal if their dot product is zero. For two vectors, u and v, this can be represented as: u ⋅ v = 0 We are given a vector, \langle 0,1,1\rangle, and we need to find two vectors that are orthogonal to it and to each other.

Define Vector Variables

Let's define two vectors, u and v, to represent our unknown vectors: u = \langle a, b, c\rangle v = \langle x, y, z\rangle We need to find real values for a, b, c, x, y, and z such that u and v are orthogonal to the given vector, and also to each other.

Write the Orthogonality Equations

For the given vector, g, and our unknown vectors, u and v, we have the following orthogonality equations: g ⋅ u = 0 g ⋅ v = 0 u ⋅ v = 0 Substituting the vector components, we get: (0, 1, 1) ⋅ (a, b, c) = 0 (0, 1, 1) ⋅ (x, y, z) = 0 (a, b, c) ⋅ (x, y, z) = 0 Expanding the dot products, we have: 1b + 1c = 0 1y + 1z = 0 ax + by + cz = 0

Solve the Equations for the Vector Components

From the first equation, we can solve b in terms of c (or vice versa): b = -c From the second equation, we can solve y in terms of z (or vice versa): y = -z Now, we can use the third equation to find two sets of values for a, b, c, x, y, and z by considering two cases: - Case 1: a = 1 - Case 2: x = 1 For both cases, let one of the unknowns have value 1. This ensures that our vectors are non-zero and thus valid.

Case 1: a = 1

If a = 1, from the third equation, ax + by + cz = 0 becomes: 1x - cz + b(-z) = 0 Let c = 1, then b = -1 (from b=-c) and substituting we get: x - z - (-z) = x = 0 So, the vector u = \langle 1, -1, 1\rangle. Now we find v by considering an orthogonal vector to u which also satisfies the second orthogonality equation: (1, -1, 1) ⋅ (x, y, z) = 0 (0, 1, 1) ⋅ (x, y, z) = 0 Expanding the dot products gives: x - y + z = 0 y + z = 0 Let x = 1, then y = -\frac{1}{2}, and z = -\frac{1}{2} (from y + z = 0). So, the vector v = \langle 1, -\frac{1}{2}, -\frac{1}{2}\rangle.

Final Answer

The two vectors that are orthogonal to the given vector, \langle 0,1,1\rangle, and to each other are: u = \langle 1, -1, 1\rangle v = \langle 1, -\frac{1}{2}, -\frac{1}{2}\rangle

Key concepts

Vector Orthogonality

Understanding vector orthogonality is crucial in solving problems in linear algebra. Two vectors are orthogonal if their dot product equals zero. This means they are at a 90-degree angle to each other. It's like saying they don't pull in the same direction at all.
For instance, suppose we have two vectors \( u \) and \( v \). They are orthogonal if \( u \cdot v = 0 \). Think of it as trying to move in two completely different directions. If one vector points north and the other points east, they won't affect each other's path.
When dealing with vector orthogonality, it often involves finding vectors that meet specific conditions, such as being orthogonal to a given vector or even to each other. This often involves solving systems of equations to ensure the dot product condition is met.

Dot Product

The dot product is a core operation in linear algebra, used to determine many relationships between vectors like length and angle. Mathematically, the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is the sum of the products of their corresponding components: \[ a_1 b_1 + a_2 b_2 + a_3 b_3. \]
If the result of this sum is zero, the vectors are orthogonal. For example, if we have a vector \( g = \langle 0, 1, 1 \rangle \) and another vector \( u = \langle a, b, c \rangle \), their dot product is \( 0 \times a + 1 \times b + 1 \times c \). Solving \( b + c = 0 \) allows us to find orthogonal vectors.
This property is widely used in computer graphics and machine learning, where finding orthogonal vectors can simplify models or help in understanding data orientations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is a fundamental part of understanding how vectors interact in multi-dimensional space.
One key aspect of linear algebra is solving systems of linear equations, often through methods like substitution or using matrix operations. The process of finding vectors orthogonal to each other involves such techniques, ensuring that their mathematical dot products satisfy the zero condition, making them orthogonal.
This branch of mathematics is instrumental in fields like physics, engineering, and computer graphics, where understanding vector spaces enables more efficient modeling and problem-solving. In the exercise, linear algebra principles helped us determine that the vectors \( u = \langle 1, -1, 1 \rangle \) and \( v = \langle 1, -\frac{1}{2}, -\frac{1}{2} \rangle \) are indeed orthogonal to \( \langle 0, 1, 1 \rangle \) and each other.