Problem 23 - 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. $$\mathbf{u}-\mathbf{v}$$

Step-by-Step Solution

Answer

\(\mathbf{u}- \mathbf{v}= 5\mathbf{i} - 12\mathbf{j}\)

Step 1: Identify the 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 difference between these two vectors \(\mathbf{u}-\mathbf{v}\).

Step 2: Subtract the \(\mathbf{i}\) components

Subtract the \(\mathbf{i}\) component of vector \(\mathbf{v}\) from the \(\mathbf{i}\) component of vector \(\mathbf{u}\). Hence, \((2 \mathbf{i}) - (-3 \mathbf{i}) = 2\mathbf{i} + 3\mathbf{i} = 5\mathbf{i}\).

Step 3: Subtract the \(\mathbf{j}\) components

Subtract the \(\mathbf{j}\) component of vector \(\mathbf{v}\) from the \(\mathbf{j}\) component of vector \(\mathbf{u}\). Hence, \((-5 \mathbf{j}) - (7 \mathbf{j}) = -5\mathbf{j} - 7\mathbf{j} = -12\mathbf{j}\).

Step 4: Combine the results

When the results from step 2 and 3 are combined, we get the vector \(\mathbf{u}-\mathbf{v} = 5\mathbf{i} - 12\mathbf{j}\).

Key Concepts

Vector Components

When working with vectors, it's crucial to understand that they are often broken down into their vector components. Vector components refer to the individual parts of the vector that represent its influence in various directions. For a two-dimensional vector like \(\mathbf{u} = 2\mathbf{i} - 5\mathbf{j}\), the components are:

• The \(\mathbf{i}\) component, which shows how far the vector reaches in the horizontal direction.
• The \(\mathbf{j}\) component, depicting the extent of the vector's influence in the vertical direction.

Understanding vector components is an essential part of performing vector arithmetic, such as vector subtraction. When subtracting vectors, you deal with their corresponding components independently. Thus, if you have another vector like \(\mathbf{v} = -3\mathbf{i} + 7\mathbf{j}\), and you want to find \(\mathbf{u} - \mathbf{v}\), you simply subtract the \(\mathbf{i}\) and \(\mathbf{j}\) components individually. This process is covered in detail through vector algebra.

Algebraic Vectors

Algebraic vectors are expressions of vectors using algebraic notation. This helps simplify calculations and clarifies manipulation of vectors with equations. A typical algebraic vector expression might look like \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) in a two-dimensional space.

• \(a\) represents the component along the x-axis.
• \(b\) signifies the component along the y-axis.

With algebraic vectors, performing operations like addition and subtraction becomes straightforward. For example, subtracting \(\mathbf{v}\) from \(\mathbf{u}\):

• Subtract \(a\) values to find the new \(\mathbf{i}\) component.
• Subtract \(b\) values to obtain the new \(\mathbf{j}\) component.

Such operations rely on combining or subtracting similar terms, adhering to basic algebraic rules. This simplification is immensely useful in solving problems involving vectors.

Two-Dimensional Vectors

Vectors in two dimensions are an excellent introduction to the concept of direction and magnitude. These vectors reside in a flat plane, represented using two components: the horizontal \(\mathbf{i}\) and vertical \(\mathbf{j}\). Each two-dimensional vector can be thought of as an arrow pointing from an origin to a specific point on this plane.
Key points about two-dimensional vectors:

• They offer a visual way to understand movements or displacements in different directions.
• Each vector has a magnitude, calculated using Pythagorean theorem: \(\sqrt{a^2 + b^2}\) for vector \(a\mathbf{i} + b\mathbf{j}\).
• The direction is often given in angles, which can be determined using trigonometric functions.

Applying operations like vector subtraction in two dimensions helps illustrate these properties by analyzing how the vectors change along their components, effectively managing movements within the 2D space. Understanding these foundational principles unlocks further comprehension of complex vector operations.