Problem 25 - Algebra and Trigonometry

Question

In Exercises \(21-38,\) let $$ \mathbf{u}=2 \mathbf{i}-5 \mathbf{j}, \mathbf{v}=-3 \mathbf{i}+7 \mathbf{j}, \text { and } \mathbf{w}=-\mathbf{i}-6 \mathbf{j} $$ Find each specified vector or scalar. $$ \mathbf{v}-\mathbf{u} $$

Step-by-Step Solution

Answer

The vector \(\mathbf{v} - \mathbf{u}\) is \(-5 \mathbf{i} + 12 \mathbf{j}\).

Step 1: Identify the initial vectors

The vectors given in the problem are \(\mathbf{u}=2 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{v}=-3 \mathbf{i}+7 \mathbf{j}\). The task is to find the vector \(\mathbf{v}-\mathbf{u}\).

Step 2: Subtraction of the i components

Subtract the \(\mathbf{i}\) component of vector \(\mathbf{u}\) from the \(\mathbf{i}\) component of vector \(\mathbf{v}\). This is calculated as \((-3) - 2 = -5\). So, the \(\mathbf{i}\) component of the resulting vector is \(-5\mathbf{i}\).

Step 3: Subtraction of the j components

Subtract the \(\mathbf{j}\) component of vector \(\mathbf{u}\) from the \(\mathbf{j}\) component of vector \(\mathbf{v}\). This is calculated as \(7 - (-5) = 12\). So, the \(\mathbf{j}\) component of the resulting vector is \(12\mathbf{j}\).

Step 4: Combine the components

Combine the \(\mathbf{i}\) and \(\mathbf{j}\) components to form the resulting vector. The resulting vector \(\mathbf{v}-\mathbf{u}\) is \(-5\mathbf{i} + 12\mathbf{j}\).

Key Concepts

Vectors

Vectors are fundamental entities in mathematics and physics, often represented as arrows pointing in a specific direction. They have both magnitude and direction, making them distinct from scalars, which have only magnitude.

In practical scenarios, vectors are commonly used to describe quantities like velocity, force, and displacement. Understanding vectors is important because they allow you to solve problems related to these quantities more effectively.

To visualize a vector, imagine an arrow:

• The length of the arrow indicates the vector's magnitude.
• The direction in which the arrow points indicates the vector's direction.

Components of Vectors

Breaking down vectors into components simplifies calculations, especially when dealing with vector arithmetic like addition or subtraction.

Every vector in a two-dimensional space can be described using its horizontal ( **x** -axis) and vertical ( **y** -axis) components. These components are often called the **i** (horizontal) and **j** (vertical) components, respectively.

When a vector is broken down into components:

• The **i-component** represents the magnitude and direction of the part of the vector that aligns with the x-axis.
• The **j-component** represents the magnitude and direction of the part of the vector that aligns with the y-axis.

For instance, if you have vector **u** = 2 **i** - 5 **j** , the 2 represents the horizontal movement and -5 the vertical movement.

Understanding components allows you to easily manipulate vectors using basic arithmetic, making problems much easier to solve.

i and j Notation

The **i** and **j** notation offers a concise way to express vectors, especially when working within two-dimensional spaces. This notation is pivotal in vector mathematics because it provides clarity and simplicity.

• **i** is the unit vector in the horizontal direction, aligning with the x-axis.
• **j** is the unit vector in the vertical direction, aligning with the y-axis.

Each unit vector has a length of one and no directionality except along the axis it represents. So, when vectors are expressed in terms of i and j, such as **v** = -3 **i** + 7 **j** , the numbers indicate how far the vector stretches along the x and y axes, respectively.

Using i and j notation simplifies the addition, subtraction, and decomposition of vectors, allowing for straightforward computation and visualization.