Problem 54 - Precalculus with Limits

Question

Writing a Linear Combination of Unit Vectors Exercises \(53-56\) , the initial and terminal points of a vector are given. Write the vector as a linear combination \(\begin{array}{ll}{\text { Initial Point }} & {\text { Terminal Point }} \\\ {(0,-2)} & {(3,6)}\end{array}\)

Step-by-Step Solution

Answer

The vector is expressed as a linear combination of unit vectors as \(3i + 8j\).

Step 1: Determine the Vector

First, calculate the vector by subtracting the initial point from the terminal point. The vector will have two components - one for the x-direction and one for the y-direction. The x-component is calculated as \(x_{\text{terminal}} - x_{\text{initial}} = 3 - 0 = 3\).The y-component is calculated as \(y_{\text{terminal}} - y_{\text{initial}} = 6 - (-2) = 8\).Thus, the vector is \((3,8)\).

Step 2: Express the Vector as a Linear Combination of Unit Vectors

Next, express the vector as a linear combination of the unit vectors \(i\) and \(j\), where \(i\) is the unit vector in the x-direction and \(j\) is the unit vector in the y-direction.This is done by multiplying the components of the vector by their corresponding unit vectors:\(3i + 8j\).This is the vector expressed as a linear combination of unit vectors.

Key Concepts

Understanding Vector Components

Vectors are fundamental objects in mathematics that describe quantity possessing both magnitude and direction. When we deal with vectors in a Cartesian coordinate system, they can be broken down into vector components—horizontal (x-direction) and vertical (y-direction) pieces—representing the effect of the vector along each coordinate axis.

Let's consider an example. If a person is walking three steps forward and eight steps upward, we can visualize this movement as a vector with an x-component representing forward steps and a y-component representing upward steps. Mathematically, this movement vector would be \( (3, 8) \), with '3' being the number of steps in the x-direction and '8' the number of steps in the y-direction.

The process of identifying these components involves comparing the vector's initial and terminal points—essentially, figuring out how much the object has moved along each axis from its starting point to its ending point. This is precisely what we do when working with vectors in precalculus.

Demystifying the Unit Vector

In the world of vectors, a unit vector plays the role of a reference or a basis, serving as a standard marker for direction with a magnitude of exactly one. Imagine the unit vector as a signpost that points the way along a specific axis in space, without specifying a distance—like saying 'that way' without stating 'how far.'

Common unit vectors are denoted by \( i \) and \( j \) in two-dimensional space. Here, \( i \) points along the positive x-axis while \( j \) points up the positive y-axis. If you want to scale the direction indicated by these unit vectors, you simply multiply them by the magnitude in their respective directions, bridging the gap between both the direction and distance that the vector represents in the coordinate system.

Vector Subtraction Simplified

Subtracting vectors is an essential process to understand movements or changes in position. Vector subtraction is visually like taking one step back for every step forward taken, or simply calculating the net movement.

In the context of the exercise, we calculate the difference in position—our vector—by subtracting the initial point from the terminal point. For instance, if the terminal point is \( (3, 6) \) and the origin or the initial point is \( (0, -2) \), we find the resultant vector by carrying out \( (3 - 0, 6 - (-2)) = (3, 8) \). This essentially answers the question, 'How far and in which direction did we move?'

The subtraction essentially removes the effect of the initial position so that the result only reflects the change, not the starting point—which is crucial in defining movement or force vectors in physics as well as in more abstract mathematical contexts.

The Relevance of Precalculus

Precalculus is the study that sets the stage for understanding calculus, and it involves a comprehensive set of mathematical skills among which is the manipulation of vectors. It bridges the gap between algebra and calculus, extending concepts such as functions, equations, and series towards more complex real-world applications, like those in physics, engineering, and computer graphics.

A crucial aspect of precalculus is learning how to deal with geometric representations of numerical relationships, just as we do with vectors. By mastering vector operations—such as addition, subtraction, and scalar multiplication—students gain competency in handling direction and magnitude, which are imperative for higher-level studies in mathematics and sciences.

Application exercises, like expressing vectors as linear combinations using unit vectors, not only solidify understanding of these concepts but also demonstrate their practical use in solving real-world problems.