Problem 4 - Calculus for Biology and Medicine

Question

Let \(A=(-1,0)\) and \(B=(2,-3)\). Find the vector representation of \(\overrightarrow{A B}\).

Step-by-Step Solution

Answer

The vector \(\overrightarrow{AB}\) is \((3, -3)\).

Step 1: Understanding the Concept

The vector \(\overrightarrow{AB}\) represents the direction and distance from point \(A\) to point \(B\). To find this vector, we subtract the coordinates of point \(A\) from the coordinates of point \(B\).

Step 2: Subtracting the X-Coordinates

Take the x-coordinates of point \(B\) and point \(A\). The calculation is as follows: \(2 - (-1) = 2 + 1 = 3\).

Step 3: Subtracting the Y-Coordinates

Take the y-coordinates of point \(B\) and point \(A\). The calculation is as follows: \(-3 - 0 = -3\).

Step 4: Forming the Vector

After subtracting the coordinates, the vector \(\overrightarrow{AB}\) is \((3, -3)\). This represents a change of 3 units in the x-direction and -3 units in the y-direction.

Key Concepts

Coordinate Subtraction

Coordinate subtraction is a straightforward yet fundamental concept in vector mathematics. To understand this process, imagine each point in a vector as having its own unique address made up of x and y coordinates. These coordinates give precise, ordered pairs of numbers that describe the location of a point in a plane.

When calculating vectors, we often need to know how to move from one point to another. To do this, we use coordinate subtraction, which involves subtracting the coordinates of the initial point from those of the endpoint, forming a direction.

• X-coordinate subtraction: Subtract the x-coordinate of the initial point from the x-coordinate of the endpoint. In our example, the calculation was: \[ 2 - (-1) = 3 \].
• Y-coordinate subtraction: Subtract the y-coordinate of the initial point from the y-coordinate of the endpoint. In the exercise, this was: \[-3 - 0 = -3 \].

Coordinate subtraction helps you find the vector \[ \overrightarrow{AB} \] that indicates the path of your movement on the plane.

Direction and Magnitude

In vector mathematics, both direction and magnitude are essential to fully understand what's happening as a vector moves from one point to another.

The **direction** of a vector is determined by the path it takes from the starting point to the endpoint. This is crucial for knowing where the vector 'points'. For the given vector \[ \overrightarrow{AB} \], the movement is represented as moving 3 units right (positive x-direction) and 3 units down (negative y-direction).

**Magnitude** is the strength or length of the vector. It’s like measuring how far two points are apart, providing a sense of scale. To find this, apply the Pythagorean theorem:

\[ |\overrightarrow{AB}| = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]
In our case,
\[ |\overrightarrow{AB}| = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \].

A vector's direction tells where it points, while its magnitude tells us how strong or far it is.

Vector Representation

Vector representation forms the backbone of visualizing mathematical movements from one position to another. It combines both the numerical and directional components of a vector to paint a complete picture.

Each vector is represented as an arrow from point A to point B, providing insight into both direction and magnitude. The vector \[ \overrightarrow{AB} = (3, -3) \] illustrates not only movement across the x-axis and y-axis but also captures the result of subtracting coordinates.

In coordinate systems, these arrows help distinguish between different vectors and identify their effects. It’s crucial to:

• Understand that \[ (x, y) \] pairs in vectors reflect the complete change in coordinates.
• Recognize that vectors transmission both direction and distance (magnitude).

To visualize vectors, imagine the vector stretching out like an arrow from the starting to the endpoint, encapsulating the journey between two positions with clear direction and strength.