Problem 21 - Calculus for Biology and Medicine

Question

In Problems , find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{r} -1 \\ 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The sum of the vectors is \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \). It is the zero vector at the origin.

Step 1: Write Vectors

First, represent the given vectors \( \mathbf{x} \) and \( \mathbf{y} \) in their component form. We have \( \mathbf{x} = \begin{bmatrix} 1 \ 0 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} -1 \ 0 \end{bmatrix} \).

Step 2: Add Vectors Algebraically

To find the sum \( \mathbf{x} + \mathbf{y} \), add the corresponding components of vectors \( \mathbf{x} \) and \( \mathbf{y} \). Calculate:\[\mathbf{x} + \mathbf{y} = \begin{bmatrix} 1 \ 0 \end{bmatrix} + \begin{bmatrix} -1 \ 0 \end{bmatrix} = \begin{bmatrix} 1-1 \ 0+0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\]

Step 3: Graphical Representation

Plot the vectors \( \mathbf{x} \), \( \mathbf{y} \), and their sum \( \mathbf{x} + \mathbf{y} \) in a coordinate plane. Vector \( \mathbf{x} \) starts from the origin and ends at (1,0). Vector \( \mathbf{y} \) also starts from the origin and ends at (-1,0). Their sum \( \mathbf{x} + \mathbf{y} \) is \( \begin{bmatrix} 0 \ 0 \end{bmatrix} \), which means it is a point at the origin.

Step 4: Explain the Vector Addition Graphically

When adding vectors, you can visualize it as connecting the tail of \( \mathbf{y} \) to the head of \( \mathbf{x} \). Since \( \mathbf{y} \) is \( \begin{bmatrix} -1 \ 0 \end{bmatrix} \), it cancels out the horizontal component of \( \mathbf{x} \), making the resultant vector \( \mathbf{x} + \mathbf{y} \) equal to the zero vector \( \begin{bmatrix} 0 \ 0 \end{bmatrix} \).

Key Concepts

Vector Representation

Vectors are mathematical objects that have both magnitude and direction, allowing us to describe quantities like velocity and force. Representing vectors is crucial for performing calculations such as addition and subtraction. To represent a vector, we often express it in terms of its components. Components break down the vector into its respective dimensions; for instance, a 2D vector is expressed in the form \( \begin{bmatrix} x \ y \end{bmatrix} \), where \( x \) and \( y \) are the horizontal and vertical components, respectively.

This exercise provides two vectors: \( \mathbf{x} = \begin{bmatrix} 1 \ 0 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} -1 \ 0 \end{bmatrix} \). Here:

• The vector \( \mathbf{x} \) has a magnitude of 1 in the positive x-direction.
• The vector \( \mathbf{y} \) has a magnitude of 1 in the negative x-direction.
• Both vectors lie completely along the x-axis, emphasizing only horizontal movement.

Understanding vector representation helps us not only with calculations but also with visualizing the behavior and interaction of vectors in a plane.

Graphical Method

The graphical method is a fundamental way of understanding vector addition by visually plotting and manipulating vectors on a coordinate plane. It is especially useful for students to develop intuition about vectors.

To add vectors using the graphical method:

• Draw each vector starting from a common origin point. In this exercise, both \( \mathbf{x} \) and \( \mathbf{y} \) originate from the origin.
• Visualize connecting the head of the first vector to the tail of the second vector. This method is sometimes referred to as the "tip-to-tail" approach.
• Whenever possible, use graph paper or digital tools to ensure accurate representations.

In the given exercise:- By plotting \( \mathbf{x} \) first, ending at point (1,0), and then adding \( \mathbf{y} \) from (1,0) back to (-1,0), we will end again at the origin, demonstrating that the vectors cancel each other out.

Resultant Vector

The resultant vector is the single vector that results from adding two or more vectors together. It represents a combined effect of all the vectors involved, like a net quantity.

To find the resultant vector algebraically, simply add the corresponding components of the vectors involved. In the exercise provided, the calculation for the vectors \( \mathbf{x} \) and \( \mathbf{y} \) led to \[\mathbf{x} + \mathbf{y} = \begin{bmatrix} 1 - 1 \ 0 + 0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\]This results in the zero vector, which lies at the origin where no net movement occurs.

• The zero vector signifies that the two vectors are equal in magnitude but opposite in direction, cancelling each other's effects.
• In physical terms, if these vectors represented forces, the object would remain stationary.

Understanding the concept of a resultant vector is vital as it helps solve real-world problems involving motion, forces, and other vector quantities efficiently.