Problem 82 - Algebra and Trigonometry

Question

If \(\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j},\) find a vector orthogonal to \(\mathbf{v}\).

Step-by-Step Solution

Answer

One vector orthogonal to \(\mathbf{v} = -2 \mathbf{i} + 5 \mathbf{j}\) is \(\mathbf{x} = 1\mathbf{i} + (2/5)\mathbf{j}\)

Step 1: Understand what it means for two vectors to be orthogonal

Two vectors are orthogonal if their dot product equals zero. The dot product \(\mathbf{a}\cdot\mathbf{b}\) of vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}\) is \(a_1b_1 + a_2b_2\). This will be used to find the vector orthogonal to \(\mathbf{v}\).

Step 2: Set up an equation

Assume the unknown vector we're looking to find is \(\mathbf{x} = x\mathbf{i} + y\mathbf{j}\). Since \(\mathbf{v}\cdot\mathbf{x} = 0\), the equation we'll solve is \((-2x) + (5y) = 0\))

Step 3: Solve the equation

This equation needs to have a nonzero solution for \((x, y)\). We can pick any value for \(x\) or \(y\) that satisfies the equation. If we let \(x = 1\), then the equation becomes \(-2 + 5y = 0\), so \(y = 2/5\). Therefore, one solution for the orthogonal vector \(\mathbf{x}\) to \(\mathbf{v}\) is \(1\mathbf{i} + (2/5)\mathbf{j}\).

Key Concepts

Dot Product

In vector mathematics, the dot product is an essential operation that allows us to compute a single number from two vectors. This resulting number encapsulates the notion of how "aligned" the two vectors are with each other.
The mathematical representation of a dot product for two-dimensional vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \) is given by:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

When the dot product is zero, the vectors are said to be orthogonal. This means they are perpendicular to each other.
Want an example? Consider \( \mathbf{v} = -2 \mathbf{i} + 5 \mathbf{j} \) and an unknown vector \( \mathbf{x} = x \mathbf{i} + y \mathbf{j} \). For orthogonality, the dot product must be zero as shown: \( (-2)x + (5)y = 0 \). By finding suitable values for \( x \) and \( y \), such as \( x = 1 \) and \( y = 2/5 \), we construct an orthogonal vector.

Vectors in Mathematics

Vectors are fundamental mathematical objects used to represent quantities having both magnitude and direction. When you think about vectors, it's essential to visualize them as arrows on a plane.
These arrows not only point in a particular direction but also have lengths that denote their magnitude. For example, a vector like \( \mathbf{v} = -2 \mathbf{i} + 5 \mathbf{j} \) is represented as an arrow pointing in the direction derived from its components on the \( \mathbf{i} \) (horizontal) and \( \mathbf{j} \) (vertical) axes.

• Magnitude: This is the length of the vector, calculated using the Pythagorean theorem as \( \sqrt{(-2)^2 + 5^2} \).
• Direction: The angle or direction the vector points towards, which is given by the ratio of its components.

Understanding vectors is crucial as they form the foundation for operations like the dot product, a key step in checking orthogonality.

Solving Equations

Solving equations involving vectors isn't drastically different from solving typical algebraic equations. The goal is to determine values that satisfy the given mathematical relationships. When we have vectors, this often involves finding components that lead an equation to a specific result.
In our example with looking for an orthogonal vector to \( \mathbf{v} = -2 \mathbf{i} + 5 \mathbf{j} \), the task is to solve \( (-2)x + (5)y = 0 \) given the vector \( \mathbf{x} = x \mathbf{i} + y \mathbf{j} \). The intuition is:

• Pick any arbitrary value for one of the unknowns (either \(x\) or \(y\)).
• Substitute this value into the equation to solve for the other unknown.

For instance, if \( x = 1 \), substituting it gives the equation \( -2 + 5y = 0 \). Solving for \( y \) gives \( y = 2/5 \). This balance effectively gives us the orthogonal vector.

Vector Components

Every vector can be broken down into its components, which are projections onto the coordinate axes. These components are crucial in performing vector arithmetic tasks and understanding the vector's behavior in systems.

• Horizontal Component: This lies along the \( \mathbf{i} \) axis and gives the vector's extent in the horizontal direction.
• Vertical Component: This is along the \( \mathbf{j} \) axis representing the vector's extent in the vertical.

Consider the vector \( \mathbf{v} = -2 \mathbf{i} + 5 \mathbf{j} \):

• Horizontal component = \(-2\)
• Vertical component = \(5\)

These components help in visualizing and computing vector-based operations like the dot product. They simplify complex vector operations by focusing on linear algebra arithmetic.