Problem 30 - Calculus for Biology and Medicine

Question

Let \(\mathbf{x}=[2,0,-1]^{\prime}\). Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

Step-by-Step Solution

Answer

The vector \( \mathbf{y} = [1, 0, 2]^{\prime} \) is one possible solution.

Step 1: Understand Perpendicular Vectors

Two vectors \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular if their dot product equals zero. Mathematically, this is expressed as \( \mathbf{x} \cdot \mathbf{y} = 0 \). In this case, \( \mathbf{x} = [2,0,-1]^{\prime} \). We'll find \(\mathbf{y} = [y_1, y_2, y_3]^{\prime}\) such that \(\mathbf{x} \cdot \mathbf{y} = 0\).

Step 2: Compute the Dot Product

The dot product \(\mathbf{x} \cdot \mathbf{y}\) is calculated as \(2y_1 + 0y_2 - 1y_3 = 0\). Simplifying this, we have the equation \(2y_1 - y_3 = 0\).

Step 3: Solve for Variables

From the equation \(2y_1 - y_3 = 0\), we can express \(y_3\) in terms of \(y_1\) as \(y_3 = 2y_1\). To find a specific \(\mathbf{y}\), we can choose a simple value for \(y_1\), such as \(y_1 = 1\), which gives \(y_3 = 2\).

Step 4: Provide General Solution

The vector \(\mathbf{y} = [y_1, y_2, 2y_1]^{\prime}\) is a solution for any real numbers \(y_1\) and \(y_2\), as the vector will still satisfy the condition \(2y_1 - y_3 = 0\) by construction. We chose \(y_1 = 1\) and \(y_2 = 0\) for simplicity, resulting in \(\mathbf{y} = [1, 0, 2]^{\prime}\).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It is used to multiply two vectors and results in a scalar value. This operation provides a way to measure how much one vector extends in the direction of another.

To compute the dot product of two vectors \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \), you multiply corresponding components and sum the results:

• First, multiply the first components: \( a_1 \times b_1 \)
• Second, multiply the second components: \( a_2 \times b_2 \)
• Third, multiply the third components: \( a_3 \times b_3 \)
• Finally, add all these products together: \( a_1b_1 + a_2b_2 + a_3b_3 \)

If the dot product is zero, it indicates that the vectors are perpendicular (orthogonal). This is the key property we used to solve the given exercise, ensuring the vectors \( \mathbf{x} = [2, 0, -1]^{\prime} \) and \( \mathbf{y} \) are perpendicular.

Vector Algebra

Vector algebra is a crucial part of mathematics that deals with vectors, objects that have both magnitude and direction. Vectors are often represented in coordinate form as sequences of numbers, like \( [x, y, z] \).

In the context of this exercise, we are using vector algebra to manage and manipulate vectors so they satisfy specific conditions, such as perpendicularity. Recall that vectors can be used not only for position and navigation in space, but for various applications such as physics calculations.

Key operations in vector algebra include:

• Vector Addition: Combining two vectors to create a new vector by adding their corresponding components.
• Scalar Multiplication: Multiplying a vector by a scalar, scaling its magnitude without changing its direction.
• Dot Product: As previously discussed, measuring the extent of one vector along the other.
• Cross Product: Operation valid in three dimensions that results in a vector perpendicular to the plane of the initial vectors.

By understanding these operations, we can handle a wide range of problems across disciplines.

Math Problem Solving

Approaching math problems successfully requires logical reasoning and analytical thinking. When dealing with vectors, such as the problem of finding a perpendicular vector, it's crucial to break down the problem into manageable steps.

In this particular exercise, we started with a clear goal: find a vector \( \mathbf{y} \) that is perpendicular to a given vector \( \mathbf{x} = [2, 0, -1]^{\prime} \). We utilized the concept of the dot product, knowing it must equal zero for perpendicular vectors.

• Comprehend the Problem: Understand what is being asked, here it was about perpendicular vectors.
• Apply Known Principles: Use vector operations like the dot product to establish equations.
• Solve Step-by-Step: Solve for unknowns by simplifying the derived equations, as seen in finding \( 2y_1 - y_3 = 0 \).
• Generalize: Recognize that there are infinitely many solutions by using parameters (like any real number for \( y_1 \)), providing a general solution family.

This methodical approach simplifies complex mathematical problems, making them more approachable for students.