Problem 45 - Advanced Engineering Mathematics

Question

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\|\mathbf{a}+\mathbf{c}\|\)

Step-by-Step Solution

Answer

\( \|\mathbf{a} + \mathbf{c}\| = \sqrt{139} \)

Step 1: Add Vectors

First, we need to find the resultant vector from adding vectors \( \mathbf{a} \) and \( \mathbf{c} \). The vector addition is performed component-wise: \( \mathbf{a} + \mathbf{c} = \langle 1,-3,2 \rangle + \langle 2,6,9 \rangle = \langle 1+2, -3+6, 2+9 \rangle = \langle 3, 3, 11 \rangle \).

Step 2: Calculate Magnitude of Vector

Now, find the magnitude of the resulting vector \( \mathbf{a} + \mathbf{c} = \langle 3, 3, 11 \rangle \). The magnitude can be calculated using the formula \( \|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2} \), where \( \mathbf{v} = \langle x, y, z \rangle \). Thus, the magnitude is \( \| \mathbf{a} + \mathbf{c} \| = \sqrt{3^2 + 3^2 + 11^2} = \sqrt{9 + 9 + 121} = \sqrt{139} \).

Key Concepts

Magnitude of a Vector

The magnitude of a vector is essentially the "length" or "size" of the vector. Imagine the vector as an arrow pointing from one point to another in space. The magnitude is like measuring the straight-line distance between the start and end of this arrow. For a vector \(\mathbf{v} = \langle x, y, z \rangle\), its magnitude is calculated using the formula: \[\|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2}\]This formula simply applies the Pythagorean theorem in three dimensions.

• The magnitude is always a non-negative number.
• If all the components of a vector are zero, its magnitude is zero, representing a point rather than a direction or length.

Understanding the magnitude helps in determining how "big" or "strong" a vector is, regardless of its direction.

Vector Operations

Vector operations include actions such as addition, subtraction, and scaling. These operations are essential for controlling and manipulating vectors in mathematical space. For example, adding two vectors is a straightforward process.Let's break down vector addition:

• Component-wise Addition: Each corresponding component of the vectors is added together. If \(\mathbf{a} = \langle x_1, y_1, z_1 \rangle\) and \(\mathbf{b} = \langle x_2, y_2, z_2 \rangle\), then their sum \(\mathbf{a} + \mathbf{b}\) is \(\langle x_1 + x_2, y_1 + y_2, z_1 + z_2 \rangle\).
• Subtraction: Similar to addition but subtract corresponding components.
• Scaling: Multiply each component by a scalar (a constant number) to change the size of the vector.

These operations are used frequently in physics and engineering to represent forces, velocities, and other directional quantities.

Euclidean Space

Euclidean space is a mathematical concept that is used to model our physical world. It is named after the ancient Greek mathematician Euclid. This space is characterized by the familiar geometric principles of flat surfaces and straight lines and is typically represented in two or three dimensions.

• Two-dimensional Euclidean Space: Think of a flat plane, like a piece of paper, where any point can be expressed with coordinates (x, y).
• Three-dimensional Euclidean Space: Represents the world we live in, where any point can be defined with coordinates (x, y, z).
• In Euclidean space, distances and angles are measured using familiar tools like the "ruler" and "protractor," adhering to Euclidean geometry.

Understanding Euclidean space is important when working with vectors because it provides the framework within which vectors operate. It allows us to apply vector operations and calculate magnitudes in a way that aligns with our real-world intuition.