Problem 22 - 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}$$$ \mathbf{x}=\left[\begin{array}{l} -3 \\ -1 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{r} -2 \\ 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The vector sum is \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} -5 \\ 2 \end{bmatrix} \). Display these using the tip-to-tail method.

Step 1: Understand the Problem

We are given vectors \( \mathbf{x} = \begin{bmatrix}-3 \ -1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix}-2 \ 3 \end{bmatrix} \). Our task is to find the sum \( \mathbf{x} + \mathbf{y} \) and represent the vectors on a plane to visualize the addition.

Step 2: Calculate the Vector Sum

To find \( \mathbf{x} + \mathbf{y} \), we add the corresponding components of \( \mathbf{x} \) and \( \mathbf{y} \). This is done as follows:\[\mathbf{x} + \mathbf{y} = \begin{bmatrix}-3 \ -1 \end{bmatrix} + \begin{bmatrix}-2 \ 3 \end{bmatrix} = \begin{bmatrix} -3 + (-2) \ -1 + 3 \end{bmatrix} = \begin{bmatrix} -5 \ 2 \end{bmatrix}\]

Step 3: Plot the Vectors

To represent each vector on a plane, treat each one as a directed line segment starting from the origin.- \( \mathbf{x} = \begin{bmatrix}-3 \ -1 \end{bmatrix} \) points 3 units left and 1 unit down.- \( \mathbf{y} = \begin{bmatrix}-2 \ 3 \end{bmatrix} \) points 2 units left and 3 units up.- \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} -5 \ 2 \end{bmatrix} \) points 5 units left and 2 units up.

Step 4: Understand Graphical Vector Addition

To graphically add \( \mathbf{x} \) and \( \mathbf{y} \), place the tail of \( \mathbf{y} \) at the head of \( \mathbf{x} \). The vector \( \mathbf{x} + \mathbf{y} \) is then the vector from the tail of \( \mathbf{x} \) to the head of \( \mathbf{y} \). This is also known as the "tip-to-tail" method.

Key Concepts

Graphical Representation of Vectors

Vectors are often represented graphically as arrows on a coordinate plane. Each vector has both direction and magnitude. The direction of a vector is given by the angle it makes with the x-axis, while its magnitude is the length of the arrow. - **Origin and components**: When drawing vectors, we usually start from the origin. The vector can be represented by the coordinates it ends at on the plane.- **Positive and negative directions**: Movement to the right and upwards is typically positive, while movement to the left and downwards is negative. For example, the vector \( \begin{bmatrix}-3 \ -1 \end{bmatrix} \) points three units left and one unit down.By graphically representing vectors, you can easily visualize their addition. Imagine each vector as a separate arrow, where the length and direction properly reflect the values of the vector components.

Two-Dimensional Vectors

Two-dimensional vectors are vectors that exist within a plane and have two components. These components correspond to the x and y axes. - **Notation**: Vectors are often denoted in the format \( \begin{bmatrix} x \ y \end{bmatrix} \). Here, \( x \) represents the horizontal component (left or right movement), and \( y \) represents the vertical component (up or down movement).- **Example**: For the vector \( \mathbf{y} = \begin{bmatrix}-2 \ 3 \end{bmatrix} \), it indicates that there is a displacement of two units to the left and three units upwards.This representation helps in breaking down complex directional movements into simple, manageable steps. Two-dimensional vectors are fundamental in physics and engineering for representing forces, velocities, and more.

Tip-to-Tail Method

The tip-to-tail method is a graphical technique used to find the resultant vector when adding two or more vectors. It simplifies the process of vector addition. Here's how you do it:- **Aligning vectors**: Start by placing the tail of the second vector at the tip (or head) of the first vector. This arrangement continues with any additional vectors.- **Resultant vector**: The new vector, known as the resultant, begins at the tail of the first vector and ends at the tip of the last vector added.- **Visualize with example**: For example, to add \( \mathbf{x} = \begin{bmatrix}-3 \ -1 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix}-2 \ 3 \end{bmatrix} \), start by drawing \( \mathbf{x} \) on the plane. Then, draw \( \mathbf{y} \), starting at the head of \( \mathbf{x} \). The resultant vector \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} -5 \ 2 \end{bmatrix} \) can then be seen as a direct line from the tail of \( \mathbf{x} \) to the head of \( \mathbf{y} \).This method provides a clear visual understanding of vector addition, enabling you to see how different vectors contribute to the overall direction and magnitude of movement.