Problem 23 - Advanced Engineering Mathematics

Question

Find \(\mathbf{a}+(\mathbf{b}+\mathbf{c})\) for the given vectors. \(\mathbf{a}=\langle 5,1\rangle, \mathbf{b}=\langle-2,4\rangle, \mathbf{c}=\langle 3,10\rangle\)

Step-by-Step Solution

Answer

\( \mathbf{a} + (\mathbf{b} + \mathbf{c}) = \langle 6, 15 \rangle \)

Step 1: Identify the Vectors

First, take note of the given vectors: \( \mathbf{a} = \langle 5, 1 \rangle \), \( \mathbf{b} = \langle -2, 4 \rangle \), and \( \mathbf{c} = \langle 3, 10 \rangle \). We are tasked with finding \( \mathbf{a} + (\mathbf{b} + \mathbf{c}) \).

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

We start by calculating the vector sum \( \mathbf{b} + \mathbf{c} \). Add the corresponding components: \( \mathbf{b}_x + \mathbf{c}_x = -2 + 3 = 1 \) for the x-component, and \( \mathbf{b}_y + \mathbf{c}_y = 4 + 10 = 14 \) for the y-component. Thus, \( \mathbf{b} + \mathbf{c} = \langle 1, 14 \rangle \).

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

Now we add vector \( \mathbf{a} \) to the result from Step 2: \( \mathbf{a} + (\mathbf{b} + \mathbf{c}) \).Add the x-components: \( 5 + 1 = 6 \), and the y-components: \( 1 + 14 = 15 \).So, \( \mathbf{a} + (\mathbf{b} + \mathbf{c}) = \langle 6, 15 \rangle \).

Key Concepts

Vectors

Vectors are fundamental in math and physics, representing quantities that have both a magnitude and a direction. Unlike scalars, which only have magnitude (like temperature or mass), vectors give us a much richer understanding of spatial information. A vector can describe anything from a force acting on an object to the velocity of a moving car.

• A vector is visualized as an arrow pointing in a specific direction.
• The length of the arrow represents the magnitude.
• A vector can be described in many contexts such as displacement, force, velocity, etc.

Knowing how to work with vectors is essential for solving complex problems in physics and engineering. Addition, subtraction, and multiplication by a scalar are basic operations you can perform on vectors. These operations are usually straightforward and follow specific rules which mirror operations on numbers.

Vector Components

To make working with vectors easier, we break them down into components. Vector components are just projections of a vector along the axes of a coordinate system. For example, a 2D vector will have two components: one along the x-axis and another along the y-axis.

• The vector components allow us to perform calculations separately for each axis.
• They simplify vector addition, subtraction, and other operations.
• Adding vectors component-wise means adding the x-components together and the y-components together.

This kind of breakdown makes complex vector operations intuitive. For instance, when adding a vector \(\mathbf{a} = \langle 5, 1 \rangle\) to another vector \(\mathbf{b} = \langle -2, 4 \rangle\), it's easy to see how you add the x and y components separately. This component-wise addition is a key skill in working with vectors.

Vector Notation

Vector notation helps in clearly specifying vectors and distinguishing them from scalar values. Vectors are often represented in bold typeface (like \( \mathbf{a} \)) or with an arrow above the letter (like \( \vec{a} \)). Commonly, vectors use angle brackets to list their components, such as \( \langle x, y \rangle \) for two-dimensional vectors.

• Using consistent vector notation makes equations and vector operations clearer.
• The brackets contain the vector's components, representing how much the vector extends in each direction.
• Understanding this notation is crucial when reading or writing vector equations.

The importance of distinguishing vectors from scalars cannot be overstated. It ensures mathematical and conceptual clarity. As shown in the original exercise, given vectors were \(\mathbf{a} = \langle 5, 1 \rangle, \mathbf{b} = \langle -2, 4 \rangle, \mathbf{c} = \langle 3, 10 \rangle\). This notation is what allows us to perform precise vector operations like addition.