Problem 17 - Calculus for Biology and Medicine

Question

In Problems 17-22, 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 \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The sum of the vectors is \( \begin{bmatrix} 4 \\ 4 \end{bmatrix} \). Graphically, this is achieved by placing \( \mathbf{y} \) at the head of \( \mathbf{x} \), arriving at \( (4,4) \).

Step 1: Identify the Problem

We need to find the sum of two vectors, \( \mathbf{x} \) and \( \mathbf{y} \). The vectors are \( \mathbf{x} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} 3 \ 2 \end{bmatrix} \).

Step 2: Vector Addition Formula

To add two vectors \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \), use the formula \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} x_1 + y_1 \ x_2 + y_2 \end{bmatrix} \).

Step 3: Calculate Vector Sum

Using the vectors given, calculate \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 1 + 3 \ 2 + 2 \end{bmatrix} = \begin{bmatrix} 4 \ 4 \end{bmatrix} \).

Step 4: Graphical Representation of Vectors

In a coordinate plane, represent \( \mathbf{x} \) as an arrow starting from the origin \( (0, 0) \) to \( (1, 2) \), and \( \mathbf{y} \) as an arrow from the origin to \( (3, 2) \).

Step 5: Graphical Addition of Vectors

To graphically add \( \mathbf{x} \) and \( \mathbf{y} \), place the tail of \( \mathbf{y} \) at the head of \( \mathbf{x} \). The result summation vector \( \mathbf{x} + \mathbf{y} \) is represented as an arrow starting from the origin \( (0,0) \) to the endpoint \((4,4)\). This represents the resultant vector.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a fundamental concept in mathematics and is essential for understanding vector addition. This plane is a two-dimensional surface made by the intersection of a horizontal axis, usually called the x-axis, and a vertical axis, known as the y-axis. The point where these two axes intersect is the origin, labeled as (0, 0).
When working with vectors like \[ \mathbf{x} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \] and \[ \mathbf{y} = \begin{bmatrix} 3 \ 2 \end{bmatrix} \], we use the coordinate plane to specifically place these vectors in their designated positions.

• The first number in a vector, known as the x-coordinate, corresponds to a position along the x-axis.
• The second number, the y-coordinate, refers to a spot along the y-axis.

For instance, the vector \( \mathbf{x} \) can be graphically represented as an arrow starting at the origin heading towards the point (1, 2). Understanding these coordinates will help you when plotting and adding vectors on the plane.

Graphical Representation of Vectors

Graphical representation is a powerful way to visualize and understand vectors. Vectors are depicted as arrows on the coordinate plane. The direction and magnitude (length) of these arrows represent the vector's characteristics.
To graph the vector \[ \mathbf{x} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \], we draw an arrow from the origin (0,0) to the point (1,2).Similarly, for \[ \mathbf{y} = \begin{bmatrix} 3 \ 2 \end{bmatrix} \], the arrow would start at the origin and end at (3,2).

• Arrows indicate direction and length signify vector magnitude.
• Placing these vectors on a graph helps in understanding spatial relations and performing vector operations like addition.

Graphically representing vectors allows you to easily perform vector addition or compare vectors by direction and size visually.

Exploring the Resultant Vector

A resultant vector is what you get when you add two or more vectors together. It shows the combined effect of the original vectors. In our exercise, we calculated the resultant vector by adding \( \mathbf{x} \) and \( \mathbf{y} \). The formula used was
\[ \mathbf{x} + \mathbf{y} = \begin{bmatrix} 4 \ 4 \end{bmatrix} \].
This result tells us that the resulting vector moves four units in the x-direction and four units in the y-direction from the origin.

• To graphically determine the resultant vector, you can place the tail of one vector at the head of the other.
• This sequential arrangement visually demonstrates their addition.

On a coordinate plane, this resultant vector is represented as a new arrow from the origin (0,0) to the endpoint (4,4). Understanding the resultant vector is crucial in depicting how multiple vectors interact and combine to form a single vector. It is a visual and mathematical way to summarize the combined directional force and magnitude.