Question

Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Algebra inequality Show that $$\left(u_{1}+u_{2}+u_{3}\right)^{2} \leq 3\left(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\right)$$ for any real numbers \(u_{1}, u_{2},\) and \(u_{3} .\) (Hint: Use the CauchySchwarz Inequality in three dimensions with \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) and choose v in the right way.)

Short answer

In conclusion, we have proven the inequality \((u_1 + u_2 + u_3)^2 ≤ 3(u_1^2 + u_2^2 + u_3^2)\) for any real numbers \(u_1, u_2, u_3\) by choosing an appropriate vector \(\mathbf{v} = \langle 1, 1, 1 \rangle\) and applying the Cauchy-Schwarz Inequality. The key steps involved calculating the dot product and magnitudes, substituting into the Cauchy-Schwarz inequality, and squaring both sides of the inequality.

Step-by-step solution

Choosing the vector \(\mathbf{v}\)

To apply the Cauchy-Schwarz Inequality, we need to find an appropriate vector \(\mathbf{v}\). Since we want to use the result of the dot product in terms of the given inequality, we can choose the vector \(\mathbf{v} = \langle 1, 1, 1 \rangle\). This choice will help to simplify the calculations and allow us to manipulate the terms accordingly.

Applying the Cauchy-Schwarz Inequality

Now that we have chosen the vector \(\mathbf{v} = \langle 1, 1, 1 \rangle\), we can apply the Cauchy-Schwarz Inequality: \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\). The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is: $$\mathbf{u} \cdot \mathbf{v}=u_{1} \cdot 1+u_{2} \cdot 1+u_{3} \cdot 1=u_{1}+u_{2}+u_{3}$$ The magnitudes of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are: $$|\mathbf{u}|=\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}$$ $$|\mathbf{v}|=\sqrt{1^{2}+1^{2}+1^{2}}=\sqrt{3}$$

Substitute into the Cauchy-Schwarz inequality

Now, using the dot product and magnitudes calculated in Step 2, we can substitute them into the Cauchy-Schwarz Inequality: $$|u_1 + u_2 + u_3| \leq |\mathbf{u}||\mathbf{v}|$$ $$|(u_1 + u_2 + u_3)| \leq \left(\sqrt{u_1^2 + u_2^2 + u_3^2}\right)\left(\sqrt{3}\right)$$

Square both sides of the inequality

To simplify and manipulate the inequality further, we can square both sides: $$\left|(u_1 + u_2 + u_3)\right|^2 \leq 3\left(u_1^2 + u_2^2 + u_3^2\right)$$ As the left side of the inequality is a square, it must be non-negative, which means that the absolute value bars can be dropped: $$\left(u_1 + u_2 + u_3\right)^2 \leq 3\left(u_1^2 + u_2^2 + u_3^2\right)$$ Thus, we have shown that \((u_1 + u_2 + u_3)^2 ≤ 3(u_1^2 + u_2^2 + u_3^2)\) for any real numbers \(u_1, u_2, u_3\) using the Cauchy-Schwarz Inequality.

Key concepts

Dot Product

Understanding the dot product is essential for diving into vector mathematics and inequalities like the Cauchy-Schwarz Inequality. Consider two vectors \( \mathbf{u} \) and \( \mathbf{v} \), each defined by their components in a coordinate system. The dot product, symbolized by \( \mathbf{u} \cdot \mathbf{v} \), is a way to multiply these vectors to give us a scalar (a single number) as result.

The computation is done by multiplying corresponding components of each vector and then summing up those products. Formally, if \( \mathbf{u} = \langle u_1, u_2, ..., u_n \rangle \) and \( \mathbf{v} = \langle v_1, v_2, ..., v_n \rangle \), then their dot product is \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + ... + u_nv_n \).

This operation not only measures the magnitude of projections of one vector along another but also plays a key role in various inequalities and properties in vector algebra, as seen in the Cauchy-Schwarz Inequality. The geometric interpretation includes the cosine of the angle between the vectors, emphasizing direction as well as magnitude in their relationship.

Vector Magnitude

In the study of vectors, knowing how to calculate vector magnitude is just as crucial as understanding their direction. The magnitude of a vector is a measure of its 'length' and is denoted by two vertical bars on either side of the vector, like \( |\mathbf{u}| \).

For a vector \( \mathbf{u} = \langle u_1, u_2, ..., u_n \rangle \), the magnitude is found by taking the square root of the sum of squares of its components, which in mathematical notation is \( |\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + ... + u_n^2} \). It corresponds to the distance from the origin to the point defined by the vector in n-dimensional space.

The concept of vector magnitude is integral when applying the Cauchy-Schwarz Inequality, as it requires us to compare the product of the magnitudes of two vectors to their dot product, revealing insights about the vectors' relative orientation and length.

Algebraic Inequalities

When solving mathematical problems, we often encounter algebraic inequalities, which are statements that describe the relative size or order of two algebraic expressions. They are denoted by symbols such as \( < \), \( >\), \( \leq \), and \( \geq \), representing 'less than,' 'greater than,' 'less than or equal to,' and 'greater than or equal to' respectively.

Inequalities become especially interesting when they bind together variable quantities, as in the context of the Cauchy-Schwarz Inequality, where we relate the squared sum of numbers to a scalar multiple of their individual squares. The power of algebraic inequalities lies in their ability to provide bounds and establish relationships that may not be immediately obvious, aiding in the proof of theorems and in solving practical problems across different fields of study such as physics, economics, and engineering.

Vector Operations

Vector operations are fundamental to many areas of mathematics and physics, allowing us to calculate and manipulate vectors in various ways. Two basic vector operations are addition and scalar multiplication. When we add two vectors, \( \mathbf{u} \) and \( \mathbf{v} \), component-wise, the result is a new vector \( \mathbf{w} = \mathbf{u} + \mathbf{v} \), where each component of \( \mathbf{w} \) is the sum of the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \).

Scalar multiplication involves multiplying every component of a vector by a scalar (a real number), effectively scaling the vector's magnitude without changing its direction. Operations like these become building blocks for more complex activities in vector analysis, such as computing dot products or using the Cauchy-Schwarz Inequality to explore the relationship between different vectors.