Problem 24 - Advanced Engineering Mathematics

Question

Find \(\mathbf{a}+(\mathbf{b}+\mathbf{c})\) for the given vectors. \(\mathbf{a}=\langle 1,1\rangle, \mathbf{b}=\langle 4,3\rangle, \mathbf{c}=\langle 0,-2\rangle\)

Step-by-Step Solution

Answer

\(\langle 5, 2 \rangle\)

Step 1: Understand the Vectors

First, identify the given vectors: \(\mathbf{a} = \langle 1, 1 \rangle\), \(\mathbf{b} = \langle 4, 3 \rangle\), and \(\mathbf{c} = \langle 0, -2 \rangle\). We need to find \(\mathbf{a} + (\mathbf{b} + \mathbf{c})\).

Step 2: Compute \(\mathbf{b} + \mathbf{c}\)

Add the vectors \(\mathbf{b}\) and \(\mathbf{c}\) by adding their corresponding components.- For the x-components: \(4 + 0 = 4\).- For the y-components: \(3 + (-2) = 1\).Thus, \(\mathbf{b} + \mathbf{c} = \langle 4, 1 \rangle\).

Step 3: Compute \(\mathbf{a} + (\mathbf{b} + \mathbf{c})\)

Now, add the vector \(\mathbf{a}\) to the result from Step 2, \(\mathbf{b} + \mathbf{c} = \langle 4, 1 \rangle\).- For the x-components: \(1 + 4 = 5\).- For the y-components: \(1 + 1 = 2\).Thus, \(\mathbf{a} + (\mathbf{b} + \mathbf{c}) = \langle 5, 2 \rangle\).

Key Concepts

Vector Operations

Vectors are a fundamental concept in mathematics and physics, representing quantities that have both magnitude and direction. We use vector operations to manipulate vectors and derive new vectors from existing ones. Among the most common operations are addition, subtraction, and scaling by a scalar.

When we talk about adding vectors, such as in this exercise, we mean combining two or more vectors to create a resultant vector. This is a straightforward yet powerful operation, especially in fields like physics where vectors are used to represent things like velocity, force, or displacement.

Additionally, vector addition is commutative and associative. This means that the order in which vectors are added does not affect the outcome. Whether you add \( \mathbf{a} \) to \( \mathbf{b} \), or \( \mathbf{b} \) to \( \mathbf{a} \), you'll end up with the same resultant vector. Similarly, grouping orders, like adding \( \mathbf{a} \) and \( \mathbf{b} \) first, and then \( \mathbf{c} \), or first \( \mathbf{b} \) and \( \mathbf{c} \) and then adding \( \mathbf{a} \), will lead to the same result.

Component Addition

Component addition is a vital part of vector arithmetic, involving adding corresponding components of vectors. Each vector can be broken down into parts, usually referred to as components, along the x-axis, y-axis, and sometimes more axes in higher dimensions.

To perform component addition, focus on each axis separately. For the exercise at hand, the vectors \( \mathbf{a} = \langle 1, 1 \rangle \), \( \mathbf{b} = \langle 4, 3 \rangle \), and \( \mathbf{c} = \langle 0, -2 \rangle \) are in two dimensions, so they have x and y components.

The process involves:

• Adding their x-components: For vectors \( \mathbf{b} \) and \( \mathbf{c} \), that’s \( 4 + 0 = 4 \).
• Adding their y-components: Here,\( 3 + (-2) = 1 \).

Resulting in a new vector \( \langle 4, 1 \rangle \) for \( \mathbf{b} + \mathbf{c} \).

After finding \( \mathbf{b} + \mathbf{c} \), repeat the process with \( \mathbf{a} \) and the newly created vector \( \langle 4, 1 \rangle \).

• X-components: \( 1 + 4 = 5 \).
• Y-components: \( 1 + 1 = 2 \).

This gives the final vector \( \langle 5, 2 \rangle \).

Vector Arithmetic

Vector arithmetic can be understood as the math dealing with these special numerical objects called vectors. Understanding vector arithmetic is crucial for tasks such as finding the resulting vector from numerous forces acting on an object, analyzing motion in physics, and much more.

For this exercise, vector arithmetic simplifies to using basic arithmetic operations applied separately to each component of a vector. Like arithmetic with regular numbers, these operations follow certain rules and properties, providing a logical structure to work with.

When adding vectors, always ensure:

• You maintain the same number of dimensions in your vectors. If one vector is in 2D, all should be.
• You apply the addition operation component-wise, which respects the principles of arithmetic, such as commutative law \((a + b = b + a)\) and associative law \((a + (b + c) = (a + b) + c)\).

By following these arithmetic rules and processes, you can accurately and efficiently solve various problems involving vectors. The key is maintaining clarity in each step of the arithmetic for correct final results. In our case, this resulted in the vector \( \langle 5, 2 \rangle \), derived from initially complex vector interactions.