Problem 55 - Precalculus with Limits

Question

In Exercises 53-56, the initial and terminal points of a vector are given. Write a linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Initial Point - \((-6, 4)\) Terminal Point - \((0, 1)\)

Step-by-Step Solution

Answer

The linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) that represents the vector from the initial point \((-6, 4)\) to the terminal point \((0, 1)\) is \( 6\mathbf{i} - 3\mathbf{j}\)

Step 1: Understand the Standard Unit Vectors

The standard unit vectors in two-dimensional space are \(\mathbf{i}\) and \(\mathbf{j}\). The vector \(\mathbf{i}\) represents one unit of distance right along the x-axis, and \(\mathbf{j}\) represents one unit of distance up along the y-axis.

Step 2: Determine the Vector from Initial Point to Terminal Point

To find the vector that goes from the initial point to the terminal point, we simply subtract the coordinates of the initial point from the coordinates of the terminal point. Given the initial point \((-6, 4)\) and terminal point \((0, 1)\), we subtract as follows: \( (0 - (-6), 1 - 4) = (6, -3) \)

Step 3: Write the Vector as a Linear Combination of Standard Unit Vectors

The resulting vector can be written as a linear combination of \(\mathbf{i}\) and \(\mathbf{j}\) by multiplying each vector by the corresponding component of our vector. Therefore, the vector is \( 6\mathbf{i} - 3\mathbf{j}\)

Key Concepts

Standard Unit Vectors

In mathematics, especially in vector calculus, standard unit vectors are fundamental building blocks for representing other vectors. The standard unit vectors in two-dimensional space are typically denoted as \( \mathbf{i} \) and \( \mathbf{j} \). These vectors are very simple yet powerful:

• \( \mathbf{i} \) points one unit in the positive x-direction. This means it moves horizontally to the right along the x-axis.
• \( \mathbf{j} \) points one unit in the positive y-direction, moving vertically upward along the y-axis.

These unit vectors provide a convenient way to describe any vector in the plane using their components. By combining these with different coefficients, one can describe any 2D vector as a linear combination of \( \mathbf{i} \) and \( \mathbf{j} \).
For example, if a vector is expressed as \( a\mathbf{i} + b\mathbf{j} \), it can be conceptualized as moving \(a\) units right or left (depending on the sign) and \(b\) units up or down on the Cartesian plane. This breakdown into standard unit vectors makes calculations and visualizations much simpler.

Initial and Terminal Points

Vectors are geometric objects that have both direction and magnitude. To define a vector in space, we often refer to its initial point and terminal point.

• The **initial point** is where the vector starts. Think of it as the tail or origin of the vector.
• The **terminal point** is where the vector ends. This is the head of the vector, indicating where the vector is pointing toward.

When given coordinates, the process of finding the vector from these points involves simple subtraction. For instance, if the initial point is \( (x_1, y_1) \) and the terminal point is \( (x_2, y_2) \), the vector \( \mathbf{v} \) from initial to terminal point is computed via \( (x_2 - x_1, y_2 - y_1) \).
This subtraction gives the direction and magnitude of the vector through its components, effectively translating the start and end points into a vector format that can be analyzed and used in further calculations like linear combinations.

Vector Components

The concept of vector components is crucial to understanding how vectors operate in different dimensions. A vector in two-dimensional space can be broken down into its essential parts known as components.

• The **x-component** is the horizontal part, often aligned with the \( \mathbf{i} \) unit vector.
• The **y-component** is the vertical part, aligned with the \( \mathbf{j} \) unit vector.

By expressing a vector in terms of its components, \( a\mathbf{i} + b\mathbf{j} \), where \(a\) and \(b\) are the vector components, you define the vector direction and its magnitude's effect in each axis.
These components are derived by subtracting coordinates or decomposing vectors in terms of their axis effects. When you express a vector using its components as a linear combination of \( \mathbf{i} \) and \( \mathbf{j} \), you make it easier to visualize and compute operations like addition, subtraction, or scalar multiplication. For instance, the vector components \( (6, -3) \) derived from an initial point of \((-6, 4)\) and a terminal point of \((0, 1)\) can be expressed as \(6\mathbf{i} - 3\mathbf{j} \), simplifying the handling of vector properties and operations.