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Chapter 5: Problem 100

Verify the Triangle Inequality for the vectors \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u}=(1,1,1), \mathbf{v}=(0,1,-2)\)

Short Answer

Expert verified
The Triangle Inequality \(|\mathbf{u}+\mathbf{v}|\leq |\mathbf{u}|+|\mathbf{v}|\) holds for the vectors \(\mathbf{u}=(1,1,1)\) and \(\mathbf{v}=(0,1,-2)\), as \(\sqrt{6}\leq \sqrt{3}+\sqrt{5}\).

Step by step solution

01

Calculating the Magnitude of Each Vector

The magnitude of a vector \(\mathbf{a}=(a_1,a_2,a_3)\) is calculated using the formula \(|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}\). So, for \(\mathbf{u}=(1,1,1)\), the magnitude is \(|\mathbf{u}|=\sqrt{1^2+1^2+1^2}=\sqrt{3}\). For \(\mathbf{v}=(0,1,-2)\), the magnitude is \(|\mathbf{v}|=\sqrt{0^2+1^2+(-2)^2}=\sqrt{5}\).
02

Adding the Vectors

To add two vectors, add corresponding components: \(\mathbf{u}+\mathbf{v}=(1+0,1+1,1-2)=(1,2,-1)\).
03

Calculating the Magnitude of the Sum Vector

Now we calculate the magnitude of the sum vector \(\mathbf{u}+\mathbf{v}\) using the formula from Step 1. So, \(|\mathbf{u}+\mathbf{v}|=\sqrt{1^2+2^2+(-1)^2}=\sqrt{6}\).
04

Verifying the Triangle Inequality

Now compare \(|\mathbf{u}+\mathbf{v}|\) with \(|\mathbf{u}|+|\mathbf{v}|\). We find \(|\mathbf{u}+\mathbf{v}|=\sqrt{6}\) and \(|\mathbf{u}|+|\mathbf{v}|=\sqrt{3}+\sqrt{5}\). Since \(\sqrt{6}\leq \sqrt{3}+\sqrt{5}\), the Triangle Inequality holds in this case.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Magnitude
Understanding the magnitude of a vector is crucial when working with vectors in any field, including physics, engineering, and computer science. The magnitude, often referred to as the length or norm, indicates the size of the vector.

In two or three-dimensional space, the magnitude of a vector \textbf{v} is calculated using the formula:
\[|\textbf{v}| = \sqrt{v_1^2 + v_2^2 + (v_3^2)}\],
where \(v_1\), \(v_2\), and \(v_3\) are the components of the vector along the x, y, and z axes, respectively. This formula extends the Pythagorean Theorem into higher dimensions and provides a way to measure the distance from the origin (or another point) to the point defined by the vector coordinates.
Vectors in Linear Algebra
In linear algebra, vectors are not just arrows in space; they are the building blocks of a much broader mathematical system. A vector in linear algebra is a tuple of numbers that can represent points in space, forces, velocities, and more.

Vectors can be described in terms of their basic properties such as direction and magnitude. Operations like addition, subtraction, and scalar multiplication are defined on vectors, and these operations abide by certain rules that make them useful for solving linear equations and performing transformations in space.
Vector Addition
Adding vectors is a fundamental operation in linear algebra. Interestingly, vector addition adheres to the commutative and associative properties, making it a predictable process.

When you add two vectors, \textbf{a} and \textbf{b}, you simply add their corresponding components to obtain a new vector \textbf{c}:
\[\textbf{c} = \textbf{a} + \textbf{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)\].
This operation can be visualized geometrically by placing the tail of vector \textbf{b} at the head of vector \textbf{a}, forming a parallelogram with the resultant vector \textbf{c} from the tail of \textbf{a} to the head of \textbf{b}'s displaced copy.
Vector Norm
The term 'norm' in linear algebra is a way to measure the size of vectors. It extends the notion of vector magnitude to a broader context which includes vector spaces beyond the typical three-dimensional space.

The most common norm, known as the Euclidean norm (or 2-norm), is essentially the same as vector magnitude and is calculated with the formula given for magnitude. However, other types of norms exist, such as the 1-norm or the infinity norm, which are useful for different applications in numerical analysis, optimization, and computer science. All norms satisfy certain properties such as non-negativity, definiteness, homogeneity, and the Triangle Inequality.

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Most popular questions from this chapter

The table shows the sales \(y\) (in millions of dollars) for Dell Incorporated during the years 1996 to 2007 . Find the least squares regression line and the least squares cubic regression polynomial for the data. Let \(t\) represent the year, with \(t=-4\) corresponding to \(1996 .\) Which model is the better fit for the data? Why? (Source: Dell Inc.) $$\begin{aligned} &\begin{array}{l|llll} \hline \text {Year} & 1996 & 1997 & 1998 & 1999 \\ \text {Sales, } y & 7759 & 12,327 & 18,243 & 25,265 \\ \hline \text {Year} & 2000 & 2001 & 2002 & 2003 \\ \text {Sales, } y & 31,888 & 31,168 & 35,404 & 41,444 \\ \hline \end{array}\\\ &\begin{array}{l|llll} \hline \\ \hline \text {Year} & 2004 & 2005 & 2006 & 2007 \\ \text {Sales, } y & 49,205 & 55,908 & 58,200 & 61,000 \\ \hline \end{array} \end{aligned}$$

Find \(\mathbf{u} \times \mathbf{v}\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). $$\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}+\mathbf{j}-\mathbf{k}$$

Find the cross product of the unit vectors [where \(\mathbf{i}=(1,0,0), \mathbf{j}=(0,1,0), \text { and } \mathbf{k}=(0,0,1)] .\) Sketch your result. $$\mathbf{j} \times \mathbf{i}$$

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). \(f(x)=\pi-x,\) fourth order

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