Question

A vector \(\vec{v}\) is given. Give two vectors that are orthogonal to \(\vec{v}\). \(\vec{v}=\langle 1,-2,3\rangle\)

Short answer

Two orthogonal vectors to \( \vec{v} = \langle 1, -2, 3 \rangle \) are \( \langle 2, 1, 0 \rangle \) and \( \langle 3, 3, 1 \rangle \).

Step-by-step solution

Understanding Orthogonality

Two vectors are orthogonal if their dot product is zero. Given the vector \( \vec{v} = \langle 1, -2, 3 \rangle \), we need to find vectors such that their dot product with \( \vec{v} \) is zero.

Set Up Equations for Orthogonal Vectors

Let \( \vec{a} = \langle a_1, a_2, a_3 \rangle \) be such a vector. The condition for orthogonality is: \( \vec{v} \cdot \vec{a} = 1 \cdot a_1 + (-2) \cdot a_2 + 3 \cdot a_3 = 0 \).

Choose Simple Values for Two Orthogonal Vectors

For the first vector, choose \( a_1 = 2 \) and \( a_2 = 1 \), then solve \( 1 \cdot 2 + (-2) \cdot 1 + 3 \cdot a_3 = 0 \) to get \( a_3 = 0 \). So, one orthogonal vector is \( \langle 2, 1, 0 \rangle \).

Choose Another Set of Values for Second Vector

For the second vector, choose \( a_1 = 3 \) and \( a_3 = 1 \), then solve \( 1 \cdot 3 - 2 \cdot a_2 + 3 \cdot 1 = 0 \), which simplifies to \( a_2 = 3 \). Thus, another orthogonal vector is \( \langle 3, 3, 1 \rangle \).

Key concepts

Vector Dot Product

The dot product is a key operation in vector algebra. If you have two vectors, say \( \vec{u} = \langle u_1, u_2, u_3 \rangle \) and \( \vec{v} = \langle v_1, v_2, v_3 \rangle \), their dot product \( \vec{u} \cdot \vec{v} \) is calculated as follows:

• Multiply their corresponding components: \( u_1 \cdot v_1 \), \( u_2 \cdot v_2 \), \( u_3 \cdot v_3 \).
• Add these products together: \( u_1 \cdot v_1 + u_2 \cdot v_2 + u_3 \cdot v_3 \).

For instance, in the vector \( \vec{v} = \langle 1, -2, 3 \rangle \), the dot product with another vector \( \vec{a} \) is calculated as:\[ \vec{v} \cdot \vec{a} = 1 \cdot a_1 + (-2) \cdot a_2 + 3 \cdot a_3 \]Understanding the dot product is essential because it serves as the basis for determining when two vectors are orthogonal. This simple operation is foundational in both geometry and physics, where it helps quantify how aligned two vectors are.

Orthogonality Condition

Orthogonality is a crucial condition in vector mathematics, particularly when dealing with geometry in multidimensional spaces. Two vectors \( \vec{u} \) and \( \vec{v} \) are considered orthogonal if their dot product equals zero:\[ \vec{u} \cdot \vec{v} = 0 \]This condition implies that the vectors are perpendicular to each other in their respective space. For example, if you have a vector \( \vec{v} = \langle 1, -2, 3 \rangle \), to find an orthogonal vector \( \vec{a} \), you set the equation:\[ 1 \cdot a_1 + (-2) \cdot a_2 + 3 \cdot a_3 = 0 \]Solving it provides values for \( a_1, a_2, \) and \( a_3 \) that satisfy this condition. This concept of orthogonality is widely used, from solving geometrical problems to applications in data science and computer graphics.

3D Vectors

3D vectors are pivotal when working in three-dimensional space. A vector in 3D space can be described using three components, usually represented as \( \langle x, y, z \rangle \). Each component corresponds to the vector's projection onto the respective axis in a 3D coordinate system.To visualize a 3D vector, imagine an arrow pointing from the origin of a 3D space towards a point \( (x, y, z) \). The length and direction of this arrow represent the vector's magnitude and direction:

• The x-component scales the direction along the x-axis.
• The y-component scales along the y-axis.
• The z-component scales along the z-axis.

When dealing with vectors in physics, engineering, or computer graphics, understanding 3D vectors is crucial for comprehending movements, forces, and directions in three-dimensional environments. Vectors in 3D space can be processed with operations like addition, subtraction, and, crucially, the dot product to determine relationships like orthogonality.

Vector Algebra

Vector algebra is a branch of mathematics concerned with the algebraic operations performed on vectors. It includes operations such as vector addition, scalar multiplication, and the vector dot product, among others. This set of operations allows us to manipulate vectors to solve problems in various fields: 1. **Addition and Subtraction**: Combine two vectors or find the difference between them. 2. **Scalar Multiplication**: Scale a vector by multiplying it with a scalar value. 3. **Dot Product**: A measure of how parallel two vectors are, crucial for testing orthogonality. These operations form the basis of more complex computations and are used to handle vectors in various dimensions and applications. Vector algebra is essential in physics for force and velocity, in engineering for mechanics, and in computer science for graphics and machine learning algorithms. Understanding these operations lets you manipulate vectors, calculate magnitudes, find angles between vectors, and solve real-world problems. Using vector algebra, we can easily realize the condition for orthogonality by employing the dot product property and solve for unknown vector components as needed.