Problem 3 - Calculus for Biology and Medicine

Question

Let \(A=(2,3)\) and \(B=(4,1)\). Find the vector representation of \(A B\)

Step-by-Step Solution

Answer

The vector representation of \(AB\) is \((2, -2)\).

Step 1: Identify Initial Point Coordinates

Point \(A\) is given as \((2,3)\). These are the coordinates of the initial point.

Step 2: Identify Terminal Point Coordinates

Point \(B\) is given as \((4,1)\). These are the coordinates of the terminal point.

Step 3: Apply the Vector Formula

The vector representation of \(AB\) is found by subtracting the coordinates of \(A\) from \(B\): \(\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1)\).

Step 4: Substitute Values into the Formula

Using the coordinates of \(A\) and \(B\), substitute into the formula: \(\overrightarrow{AB} = (4-2, 1-3)\).

Step 5: Calculate the Vector Values

Perform the subtraction: \(\overrightarrow{AB} = (2, -2)\).

Key Concepts

Vector Mathematics

Vector mathematics is a critical component in understanding both physical phenomena and computational problems. Vectors are mathematical entities that have both magnitude and direction, distinguishing them from ordinary numbers or scalars, which only have magnitude. Vectors are often represented using the notation of an arrow or a pair of coordinates, such as the representation you encountered with points A and B.

In mathematical terms, a vector is denoted as:

• A column or row of numbers: \[ \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \]
• A bold letter, such as \(\mathbf{v}\)
• An arrow above the symbol, like \(\vec{v}\)

These notations make it clear that a distinction between direction and magnitude is being made. Vectors help us perform complex operations, such as rotations and transformations, in multidimensional spaces efficiently.

Coordinate Geometry

Coordinate Geometry, often referred to as analytic geometry, bridges algebra and geometry through the use of coordinates. Within this framework, points, lines, and shapes are represented in a numerical format that facilitates analysis and computation.

Understanding coordinates is essential for solving the exercise you were given. In a 2D space, any point can be expressed as \((x, y)\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.

Every operation in coordinate geometry, such as finding a vector between two points, relies on the principles of this system. By knowing the coordinates of two points—\(A\) as \((2, 3)\) and \(B\) as \((4, 1)\)—you were able to determine their vector representation. This is a hallmark of coordinate geometry, translating spatial concepts into solvable algebraic expressions.

Vector Subtraction

Vector subtraction is a fundamental technique used to find the vector between two points, an operation crucial for solving your exercise. To subtract vectors in a two-dimensional space, you simply subtract their corresponding components:

• Subtract the x-coordinates: \(x_2 - x_1\)
• Subtract the y-coordinates: \(y_2 - y_1\)

For your specific example, the process was applied to points \(A\left(2, 3\right)\) and \(B\left(4, 1\right)\). By taking the difference of these coordinates, you arrived at the vector representation \(\overrightarrow{AB} = \left(4 - 2, 1 - 3\right) = (2, -2)\). This means the vector from \(A\) to \(B\) points 2 units in the positive x-direction and 2 units in the negative y-direction.

This method of vector subtraction is invaluable for breaking down complex problems into manageable parts. It applies not only in geometry or physics, where direction matters but also in various engineering and computer science applications.