Problem 5 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\(\overrightarrow{A B} = (-1, -2, 5)\).

Step 1: Understand the vector representation

The vector representation \(\overrightarrow{A B}\) is a vector pointing from point \(A\) to point \(B\). To find this, we need to calculate the difference between the coordinates of \(B\) and \(A\).

Step 2: Subtract the coordinates

To find \(\overrightarrow{A B}\), subtract each coordinate of point \(A\) from the corresponding coordinate of point \(B\): \(\overrightarrow{A B} = (x_B - x_A, y_B - y_A, z_B - z_A)\).

Step 3: Calculate each component

Calculate the \(x\)-component: \(-1 - 0 = -1\). Calculate the \(y\)-component: \(-1 - 1 = -2\). Calculate the \(z\)-component: \(2 - (-3) = 2 + 3 = 5\).

Step 4: Write the vector

Now that we have each component calculated, the vector \(\overrightarrow{A B}\) is \(\overrightarrow{A B} = (-1, -2, 5)\).

Key Concepts

Vectors in Calculus

Vectors play a pivotal role in calculus, especially when dealing with functions of several variables. A vector is essentially an arrow in a space, defined by a direction and a magnitude. Think of it as a journey from one point to another, where it not only describes how far you need to go but also in what direction. In calculus, vectors help us analyze rates of change and can represent physical quantities like force or velocity.
In the context of our exercise, the vector \(\overrightarrow{A B}\), represents the journey from point \(A\) to point \(B\) in a three-dimensional space. Understanding vectors in calculus involves grasping concepts such as gradient, divergence, and curl, which use vectors to explore changes in fields. These concepts extend the idea of a slope in one dimension to more complex multi-dimensional surfaces.
Both physical and geometrical interpretations of vectors enrich our understanding in calculus, allowing a more comprehensive analysis of problems within that realm.

Coordinate Subtraction

The idea of coordinate subtraction is central to finding vector representations. Essentially, it provides a method to calculate the difference between two points in space, resulting in a vector that conveys the displacement.
When you perform coordinate subtraction, here's what happens:

• Subtract the x-coordinate of the initial point from the x-coordinate of the terminal point.
• Subtract the y-coordinate of the initial point from the y-coordinate of the terminal point.
• Subtract the z-coordinate of the initial point from the z-coordinate of the terminal point.

This process works similarly in any dimension and helps figure out how far and in what direction one point is from another. For our points \(A = (0, 1, -3)\) and \(B = (-1, -1, 2)\), subtracting corresponding coordinates helped us determine that \(\overrightarrow{A B} = (-1, -2, 5)\).
Coordinate subtraction is, therefore, the fundamental step towards understanding vector direction and magnitude, as it allows us to pinpoint the exact path from one position to another.

3D Vectors

Three-dimensional (3D) vectors are indispensable for representing spatial relationships and entities in our physical world. A 3D vector is composed of three components: an x-component, a y-component, and a z-component. These components form a tuple that represents a point's position in three-dimensional space.
In our example, the 3D vector \(\overrightarrow{A B} = (-1, -2, 5)\) captures the position change from point \(A\) to point \(B\). This vector not only tells us how to travel in the x-and y-directions but also how to move up or down in the z-direction, providing a complete picture of motion in our space.
The major applications of 3D vectors span across physics, computer graphics, and engineering, where understanding how objects interact and move in a three-dimensional setting is crucial. This notion of a 3D vector also aligns with physical phenomena like motion under gravity and electromagnetic fields, offering a robust mathematical framework to predict and analyze diverse conditions.