Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. $$\langle- 5,8\rangle$$

Step-by-Step Solution

Answer

The vector \( \langle -5, 8 \rangle \) is \(-5i + 8j\).

Step 1: Understand the vector components

Given the vector \( \langle -5, 8 \rangle \), it is composed of two components: \( -5 \) and \( 8 \). These correspond to the x-component and the y-component of the vector, respectively.

Step 2: Write in terms of i and j

Rewrite the vector \( \langle -5, 8 \rangle \) using the unit vectors \( i \) and \( j \). The x-component \( -5 \) is multiplied by the unit vector \( i \), and the y-component \( 8 \) is multiplied by the unit vector \( j \).

Step 3: Express the vector

Combine the x and y components to express the vector in the form of \( ai + bj \). Thus, the vector \( \langle -5, 8 \rangle \) can be written as \(-5i + 8j\).

Key Concepts

Vector Components

Vectors are entities that have both magnitude and direction. When dealing with vectors, a crucial concept is understanding their components. Each vector can be broken down into parts that align with the axes of a coordinate system. These parts are known as vector components. These components effectively tell us how much of the vector acts in the direction of each axis.
For instance, given a two-dimensional vector denoted by \(\langle -5, 8 \rangle\), it consists of two separate components:

• The x-component is \(-5\), which represents the vector's movement along the x-axis.
• The y-component is \(8\), which indicates the movement along the y-axis.

By understanding these components, we can easily manage and manipulate vectors using basic arithmetic and algebra.

Unit Vectors

In vector notation, unit vectors play a pivotal role. A unit vector is a vector with a magnitude of 1 that is used to express directions along a specific axis. The most common unit vectors are:

• \(\mathbf{i}\) for the x-direction.
• \(\mathbf{j}\) for the y-direction.

These unit vectors allow us to express any vector in terms of its components along the coordinate axes. For example, if a vector points completely in the x-direction with a magnitude of 3, it can be written as \(3\mathbf{i}\).
They are extremely helpful in breaking down vectors into their components by multiplying the components with these unit vectors, which maintains the vector's direction.

Vector Expression

A vector expression is a way to represent a vector using its components and unit vectors. When given a vector in the form \( \langle x, y \rangle \), it can be expressed in the notation \( ai + bj \) where \(a\) and \(b\) are the coefficients of the unit vectors.
For example, the vector \( \langle -5, 8 \rangle \) can be expressed using unit vectors as \(-5\mathbf{i} + 8\mathbf{j}\), where:

• The coefficient \(-5\) matches the x-component of the vector.
• The coefficient \(8\) corresponds to the y-component.

This method of expression is very beneficial as it explicitly shows how the vector interacts with each axis.

X-Component

The x-component of a vector represents its horizontal influence in a coordinate plane. In a vector notation \(\langle x, y \rangle\), the x-component is the first value, which indicates how far and in which direction the vector extends along the x-axis.
For the vector \( \langle -5, 8 \rangle \), the x-component is \(-5\). This indicates that the vector moves 5 units in the negative x-direction. By multiplying this component with the unit vector \(\mathbf{i}\), we get \(-5\mathbf{i}\), capturing the complete influence of this vector along the x-axis.

Y-Component

Similar to the x-component, the y-component of a vector determines its vertical influence. For a vector \(\langle x, y \rangle\), the y-component is the second value, representing movement along the y-axis.
In our specific vector example \( \langle -5, 8 \rangle \), the y-component is \(8\). This tells us that the vector moves 8 units in the positive y-direction. By associating this component with the unit vector \(\mathbf{j}\), the y-component becomes \(8\mathbf{j}\), which reflects the vector's action along the y-axis precisely.