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

\(\mathbf{y} = \begin{bmatrix} k \\ k \end{bmatrix}\) for any real number \(k\).

Step 1: Understand the Perpendicularity Condition

For two vectors \(\mathbf{x}\) and \(\mathbf{y}\) to be perpendicular, their dot product must be zero. Thus, we need: \(\mathbf{x} \cdot \mathbf{y} = 0\).

Step 2: Write the Dot Product Equation

Given \(\mathbf{x} = \begin{bmatrix} 1 \ -1 \end{bmatrix}\), assume \(\mathbf{y} = \begin{bmatrix} a \ b \end{bmatrix}\). The dot product is \(1 \cdot a + (-1) \cdot b = 0\), leading to the equation: \(a - b = 0\).

Step 3: Solve the Equation for a Relationship between a and b

From \(a - b = 0\), we can infer that \(a = b\). Thus, any vector of the form \(\begin{bmatrix} a \ a \end{bmatrix}\) will satisfy the dot product condition.

Step 4: Provide a General Solution

Since \(a = b\), \(\mathbf{y}\) can be any scalar multiple of \(\begin{bmatrix} 1 \ 1 \end{bmatrix}\). Hence, a general form for \(\mathbf{y}\) is \(\begin{bmatrix} k \ k \end{bmatrix}\), where \(k\) is any real number.

Key Concepts

Understanding the Dot Product

The dot product is a fundamental operation in vector mathematics. It involves taking two vectors and calculating a single number that tells us something about their relationship. When working with vectors \( \mathbf{x} = \begin{bmatrix} x_1 & x_2 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} y_1 & y_2 \end{bmatrix} \), the dot product is given by: \( x_1 \cdot y_1 + x_2 \cdot y_2 \). This operation is critical because it helps determine perpendicularity between vectors. If the dot product is zero, this means the vectors are perpendicular to one another, i.e., they meet at a right angle. This property is highly useful in various fields of mathematics and physics, enabling us to understand more complex concepts such as orthogonality in vector spaces. In the exercise we're considering, verifying that the dot product equals zero was key to defining the relationship between the vectors.

Linear Algebra and Its Power

Linear algebra is the branch of mathematics that deals with vectors, matrices, and systems of linear equations. It provides the language for understanding various geometric and algebraic relationships, including those involving perpendicular vectors. In this context, linear algebra allows us to determine relationships like parallelism and perpendicularity through the manipulation of equations and operations like the dot product.

Linear algebra is fundamental in solving systems of equations. This is because systems of equations often arise when trying to find relationships between multiple variables and constraints. In our exercise, linear algebra principles allowed us to derive the condition \( a = b \) from the equation \( a - b = 0 \), which was crucial for finding the vector \( \mathbf{y} \) perpendicular to \( \mathbf{x} \). This showcases linear algebra's ability to simplify complex problems into manageable steps that can be solved analytically.

Systems of Equations Solved Easily

A system of equations is essentially a set of equations with multiple variables. Solving systems of equations involves finding values for the variables that satisfy all the equations simultaneously. There are several methods to solve these systems, including substitution, elimination, and using matrices. In our scenario, we were dealing with a very simple system: the equation \( a - b = 0 \). This easily translates to \( a = b \), giving us the solution directly.

The power of systems of equations lies in their ability to handle multiple constraints and relationships between variables simultaneously. They are widely used in various real-world applications, from engineering to economics. Understanding and solving systems of equations can lead to insights on how different variables interact, as seen in our exercise where a simple system resulted in finding many possible vectors \( \mathbf{y} \) that are perpendicular to \( \mathbf{x} \). This illustrates how even simple systems can reveal profound geometrical properties in higher dimensions.