Problem 14 - Advanced Engineering Mathematics

Question

$$ \mathbf{a}=\langle 3,1\rangle+\langle-1,2\rangle, \mathbf{b}=\langle 6,5\rangle-\langle 1,2\rangle $$

Step-by-Step Solution

Answer

\(\mathbf{a} = \langle 2, 3 \rangle\), \(\mathbf{b} = \langle 5, 3 \rangle\)."

Step 1: Add Vectors for \(\mathbf{a}\)

The vector \(\mathbf{a}\) is given by the addition of two vectors: \(\langle 3,1\rangle\) and \(\langle-1,2\rangle\). To find this sum, add corresponding components. For the first component, \(3 + (-1) = 2\). For the second component, \(1 + 2 = 3\). Therefore, \(\mathbf{a} = \langle 2, 3 \rangle\).

Step 2: Subtract Vectors for \(\mathbf{b}\)

The vector \(\mathbf{b}\) is given by subtracting \(\langle 1,2 \rangle\) from \(\langle 6,5 \rangle\). Subtract the corresponding components. For the first component, \(6 - 1 = 5\). For the second component, \(5 - 2 = 3\). Therefore, \(\mathbf{b} = \langle 5, 3 \rangle\).

Key Concepts

Component-wise Operations

When dealing with vectors, one of the core practices is operating on them component-wise. This means that when you add or subtract vectors, you do so by handling each corresponding element separately. Suppose you have two vectors, \( \langle x_1, y_1 \rangle \) and \( \langle x_2, y_2 \rangle \). When performing operations, you adjust each of these elements by their respective counterparts.
For example:

• For addition: \( \langle x_1 + x_2, y_1 + y_2 \rangle \)
• For subtraction: \( \langle x_1 - x_2, y_1 - y_2 \rangle \)

Using component-wise operations ensures the integrity and consistency of the results. It preserves the structure of vector mathematics so calculations remain accurate. Breaking these vectors into their components can simplify complex numbers and helps in visualizing operations like translation in 2D space.

Vector Subtraction

Vector subtraction is often used to find the difference between two vectors or to point from one vector to another. This operation is similar to vector addition but involves taking away one vector's components from another.
Imagine two vectors, \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \). The subtraction \( \mathbf{u} - \mathbf{v} \) results in a new vector \( \langle u_1 - v_1, u_2 - v_2 \rangle \).
Here's why this is useful:

• Direction and Change: Subtracting vectors shows how far and in which direction one vector extends from another.
• Negative Vectors: When you subtract a vector, it’s equivalent to adding its negative.

For example, subtracting \( \langle 1,2 \rangle \) from \( \langle 6,5 \rangle \) results in \( \langle 5,3 \rangle \), indicating a shift or movement represented by these numbers.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations and linear functions and their representations through matrices and vector spaces. Vectors are fundamental in linear algebra and are considered as objects that allow us to express quantities that have both a magnitude and a direction.
Here are some crucial aspects:

• Vector Spaces: A collection of vectors, where vector addition and scalar multiplication are defined.
• Operations: Component-wise addition and subtraction of vectors are foundational operations in linear algebra.
• Applications: These simple operations extend to more complex tasks, like transformations in computer graphics or solving systems of equations.

In the context of linear algebra, performing operations with vectors can unravel patterns and find solutions to challenging problems, showcasing their utility across various scientific and engineering disciplines.