Problem 17 - Calculus for Biology and Medicine

Question

Find \(\mathbf{x}+\mathbf{y}\) for each pair of vectors \(\mathbf{x}\) and \(\mathbf{v}\). Represent \(\mathrm{x}, \mathbf{y}\), and \(\mathrm{x}+\mathrm{y}\) in the plane, and explain graphically how you add \(\mathbf{x}\) and \(\mathbf{y}\). \(\mathbf{x}=\left[\begin{array}{l}1 \\ 2\end{array}\right]\) and \(\mathbf{y}=\left[\begin{array}{l}3 \\ 0\end{array}\right]\)

Step-by-Step Solution

Answer

The vector sum is \( \begin{bmatrix} 4 \\ 2 \end{bmatrix} \). Graphically, it represents the endpoint when adding \( \mathbf{x} \) and \( \mathbf{y} \) starting from the origin.

Step 1: Understand Vectors

Vectors \( \mathbf{x} \) and \( \mathbf{y} \) are given as column matrices. \( \mathbf{x} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \) and \( \mathbf{y} = \begin{bmatrix} 3 \ 0 \end{bmatrix} \). They represent points or directions in a 2-dimensional space.

Step 2: Add Vectors

To find \( \mathbf{x} + \mathbf{y} \), add corresponding components of the vectors. For this problem:\[ \mathbf{x} + \mathbf{y} = \begin{bmatrix} 1 \ 2 \end{bmatrix} + \begin{bmatrix} 3 \ 0 \end{bmatrix} = \begin{bmatrix} 1 + 3 \ 2 + 0 \end{bmatrix} = \begin{bmatrix} 4 \ 2 \end{bmatrix} \].

Step 3: Represent Vectors on the Plane

We can plot these vectors in a coordinate plane with an x-axis and a y-axis. The tail of each vector starts at the origin (0, 0). Vector \( \mathbf{x}\) at (1, 2) points in the direction of 1 unit right, 2 units up. Vector \( \mathbf{y}\) at (3, 0) points in the direction of 3 units right.

Step 4: Vector Addition Graphically

Graphically, vector addition means you place the tail of one vector at the head of the other. Starting with \( \mathbf{x} \), draw \( \mathbf{y} \) starting from the endpoint of \( \mathbf{x} \). The resulting vector \( \mathbf{x} + \mathbf{y} \) is from the origin to this new endpoint. This represents the endpoint (4,2) of the vector \( \begin{bmatrix} 4 \ 2 \end{bmatrix} \) on the coordinate plane.

Key Concepts

Vectors in Two-Dimensional Space

When it comes to vectors in two-dimensional space, they play a crucial role in representing both direction and magnitude in a plane. This is akin to discussing positions or directions based on two coordinates: an x-coordinate and a y-coordinate.
A vector such as \( \mathbf{x} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \) indicates that from the origin, you move 1 unit along the x-axis, and 2 units up along the y-axis. Similarly, \( \mathbf{y} = \begin{bmatrix} 3 \ 0 \end{bmatrix} \) means moving 3 units along the x-axis without any vertical displacement. Here are a few key points:

• Vectors have both direction and magnitude.
• You can think of them as arrows pointing from the origin to a particular point in space.
• They are expressed in the form \( \begin{bmatrix} x \ y \end{bmatrix} \), representing their x and y components respectively.

Graphical Representation of Vectors

Graphs are a powerful method for visualizing vectors. By plotting vectors on a graph, you create a clear visual image of their magnitude and direction.
Each vector can be represented as an arrow starting from the origin of a coordinate plane and extending to a point dictated by its components. For instance, the vector \( \mathbf{x} = \begin{bmatrix} 1 \ 2 \end{bmatrix} \) can be depicted by drawing an arrow starting at point (0,0) and ending at point (1,2). Similarly, \( \mathbf{y} = \begin{bmatrix} 3 \ 0 \end{bmatrix} \) is shown as an arrow reaching point (3,0) from the origin.
Some helpful tips when plotting:

• Start drawing each vector from the origin, which is point (0,0).
• Ensure the length and direction of the arrow accurately reflect the magnitude and direction of the vector.
• Use a consistent scale to maintain proportional relationships between different vectors.

Graphical representation aids in understanding vector addition by visually showing how vectors connect end to end.

Coordinate Plane Representation

The coordinate plane is an essential tool in mathematics for representing vectors and conducting operations like addition. It consists of two perpendicular lines or axes: the x-axis (horizontal) and the y-axis (vertical).
In adding vectors \( \mathbf{x} \) and \( \mathbf{y} \), we plot them on this coordinate plane. The addition, \( \mathbf{x} + \mathbf{y} = \begin{bmatrix} 4 \ 2 \end{bmatrix} \), means finding a new vector that represents the sum. Here’s a quick breakdown of the process:

• Begin by plotting vector \( \mathbf{x} \), starting from the origin moving to point (1,2).
• Next, plot vector \( \mathbf{y} \) starting from the endpoint of vector \( \mathbf{x} \), reaching to point (4,2).
• The resultant vector, \( \mathbf{x} + \mathbf{y} \), will hence stretch from the origin (0,0) directly to this new point (4,2).

By utilizing the coordinate plane, you not only gain a graphical illustration of vectors but also a powerful method to compute and visualize vector addition and other operations.