Problem 27 - Calculus for Biology and Medicine

Question

Let \(\mathbf{x}=[1,1]^{\prime}\). Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

Step-by-Step Solution

Answer

\( \mathbf{y} = [1, -1]' \) is perpendicular to \( \mathbf{x} = [1, 1]' \).

Step 1: Understand Perpendicular Vectors

Two vectors are perpendicular if their dot product is zero. In this problem, the vectors \( \mathbf{x} \) and \( \mathbf{y} \) are perpendicular, meaning \( \mathbf{x} \cdot \mathbf{y} = 0 \).

Step 2: Set Up Dot Product Equation

Let \( \mathbf{y} = [y_1, y_2]' \). The dot product \( \mathbf{x} \cdot \mathbf{y} \) is computed as \( 1 \cdot y_1 + 1 \cdot y_2 = y_1 + y_2 \). Therefore, the condition for perpendicularity becomes \( y_1 + y_2 = 0 \).

Step 3: Solve for Components of \( \mathbf{y} \)

Given the equation \( y_1 + y_2 = 0 \), you can express \( y_2 \) in terms of \( y_1 \): \( y_2 = -y_1 \). Thus, any vector of the form \( \mathbf{y} = [y_1, -y_1]' \) will be perpendicular to \( \mathbf{x} = [1, 1]' \).

Step 4: Choose Specific Solution for \( \mathbf{y} \)

Select a specific value for \( y_1 \) to find a particular solution. For simplicity, let \( y_1 = 1 \), then \( y_2 = -1 \), resulting in \( \mathbf{y} = [1, -1]' \).

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It provides a method to multiply two vectors, yielding a scalar result. The dot product of two vectors \(\mathbf{a} = [a_1, a_2]'\) and \(\mathbf{b} = [b_1, b_2]'\) is calculated using the formula:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\]This calculation involves multiplying corresponding components and summing them. It serves as a useful tool to determine the relationship between vectors. One key application is identifying perpendicular vectors. Two vectors are perpendicular if their dot product equals zero. This means there is no alignment between them. In simpler terms, computing the dot product answers whether vectors point in orthogonal directions, forming a right angle between them.

Perpendicular Vectors

Perpendicular vectors, also known as orthogonal vectors, are vectors whose dot product is zero. This orthogonality implies that they form a right angle, or 90 degrees, with each other. In the context of the exercise, we determined that vectors \(\mathbf{x} = [1, 1]'\) and \(\mathbf{y} = [y_1, y_2]'\) are perpendicular if their dot product \(y_1 + y_2\) equals zero.

• Setting the dot product to zero gives the condition \(y_1 + y_2 = 0\).
• This equation implies that the sum of the vector components must cancel out to zero for perpendicularity.

Using this principle ensures we correctly identify vectors that are perpendicular. It also emphasizes the dot product’s role in defining vector relationships in spaces.

Vector Components

Vector components refer to the individual elements that comprise a vector. Each vector in a 2-dimensional space, like \(\mathbf{y} = [y_1, y_2]'\), has two components that dictate its position and direction. Understanding these components is crucial, as they determine how a vector behaves under various operations like addition, subtraction, and the dot product.In the given problem, the components of \(\mathbf{y}\) need to satisfy the equation \(y_1 + y_2 = 0\) for \(\mathbf{y}\) to be perpendicular to \(\mathbf{x} = [1, 1]'\). This relationship highlights the importance of manipulating individual vector components:

• When \(y_1\) changes, \(y_2\) must adjust accordingly to maintain the perpendicularity condition.
• We can express \(y_2\) as \(-y_1\), showing a negative relationship between the components.

By exploring vector components, we gain insight into the spatial arrangement and relationship of vectors.