Problem 3 - Calculus for Biology and Medicine

Question

Let \(A=(2,3)\) and \(B=(1,1)\). Find the vector representation of \(\overrightarrow{A B}\).

Step-by-Step Solution

Answer

The vector representation of \(\overrightarrow{AB}\) is \((-1, -2)\).

Step 1: Understand the Concept

The vector representation of a line segment \ \(\overrightarrow{AB}\ \) from point \ \(A\ \) to point \ \(B\ \) is a vector that points from \ \(A\ \) to \ \(B\ \). It is calculated by subtracting the coordinates of point \ \(A\ \) from the coordinates of point \ \(B\ \).

Step 2: Identify Coordinates

Point \ \(A\ \) has coordinates \ \((x_1, y_1) = (2,3)\ \) and point \ \(B\ \) has coordinates \ \((x_2, y_2) = (1,1)\ \).

Step 3: Subtract Coordinates

To find the vector \ \(\overrightarrow{AB}\ \), subtract the coordinates of \ \(A\ \) from \ \(B\ \): \[ x_2 - x_1 = 1 - 2 = -1 \]\[ y_2 - y_1 = 1 - 3 = -2 \]

Step 4: Write Vector Representation

Combine the differences in a vector notation: \ \(\overrightarrow{AB} = (-1, -2)\ \).

Key Concepts

Line Segment

A line segment is a straight path connecting two points, having a definite beginning and end. In geometry, line segments are commonly represented by endpoints. For instance, in our exercise, the points are labeled as \(A\) and \(B\), connected directly by the line segment \(\overrightarrow{AB}\). This means that it starts from \(A\) and ends at \(B\).
In the context of vectors, a line segment also acts as a vector. It has both direction and magnitude, allowing us to consider it as more than just a simple line.
When discussing the vector representation of a line segment, we focus on translating this simple geometric figure into a mathematical vector that can be worked with in various calculations. It's important to note:

• The direction of the vector from A to B matters.
• The length of the line segment gives us the vector's magnitude.

Coordinate Subtraction

Coordinate subtraction is a simple yet powerful technique that helps in finding the vector representation of a line segment. It involves subtracting the corresponding coordinates of two points in a very straightforward manner.
To illustrate with our exercise, we have two points A\((x_1, y_1) = (2,3)\) and B\((x_2, y_2) = (1,1)\). We want to find the vector from A to B. To do so, we simply subtract the coordinates of A from those of B.
Simplifying:

• The x-coordinate of the vector is computed as \(x_2 - x_1\), resulting in \(1 - 2 = -1\).
• The y-coordinate is calculated as \(y_2 - y_1\), resulting in \(1 - 3 = -2\).

This subtraction method provides us with the vector components that show both the direction and how far point A is from point B in both the x and y directions.

Vector Notation

Vector notation is a concise way of expressing a vector mathematically. It not only represents the vector's direction and magnitude, but also its position in space. The notation often uses angular brackets or parentheses to enclose the vector's components.
In the example of \(\overrightarrow{AB}\), the vector notation becomes \((-1, -2)\). This tells us that:

• The vector moves one unit to the left along the x-axis.
• It also moves two units down along the y-axis.

Such representations are extremely useful in interpreting vector direction and magnitude at a glance.
Using vector notation, we can easily perform calculations, especially in physics and engineering, where vectors are critical. It allows for a clear communication of the vector's properties, enabling us to easily plot or compare them on a graph.