Question

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

Short answer

Question: Prove that for any real numbers \(u_{1}, u_{2}\) and \(u_{3}\), the inequality \(\left(u_{1}+u_{2}+u_{3}\right)^{2} \leq 3\left(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\right)\) holds using the Cauchy-Schwarz Inequality. Answer: Using the Cauchy-Schwarz Inequality and choosing a suitable vector \(\mathbf{v} = \langle 1, 1, 1 \rangle\), we compute the dot product and magnitudes of the vectors. Applying the inequality, we get \((u_{1} + u_{2} + u_{3})^2 \leq 3(u_{1}^2 + u_{2}^2 + u_{3}^2)\), thus proving the given inequality.

Step-by-step solution

Apply the Cauchy-Schwarz Inequality

By the Cauchy-Schwarz Inequality, we have \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\). Now we need to choose a suitable vector \(\mathbf{v}\) that will help us prove the desired inequality.

Choose suitable vector \(\mathbf{v}\)

Let's choose the vector \(\mathbf{v} = \langle 1, 1, 1 \rangle\). This choice allows us to work with the sum of the components of \(\mathbf{u}\) in the dot product, which appears in the desired inequality.

Compute the dot product

Now, we compute the dot product \(\mathbf{u} \cdot \mathbf{v}\): \(\mathbf{u} \cdot \mathbf{v} = u_{1}(1) + u_{2}(1) + u_{3}(1) = u_{1} + u_{2} + u_{3}\).

Compute the magnitudes of the vectors

We need to find the magnitudes of the vectors \(\mathbf{u}\) and \(\mathbf{v}\): \(|\mathbf{u}| = \sqrt{u_{1}^2 + u_{2}^2 + u_{3}^2}\) and \(|\mathbf{v}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\).

Apply the Cauchy-Schwarz Inequality and simplify

Apply the Cauchy-Schwarz Inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\): \(|u_{1} + u_{2} + u_{3}| \leq \sqrt{u_{1}^2 + u_{2}^2 + u_{3}^2} \cdot \sqrt{3}\). Since all terms are non-negative or non-positive, squaring both sides of the inequality doesn't change the inequality direction: \((u_{1} + u_{2} + u_{3})^2 \leq 3(u_{1}^2 + u_{2}^2 + u_{3}^2)\). Thus, we have successfully shown that for real numbers \(u_{1}, u_{2},\) and \(u_{3}\), it is true that \(\left(u_{1}+u_{2}+u_{3}\right)^{2} \leq 3\left(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\right)\) using the Cauchy-Schwarz Inequality.

Key concepts

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors, resulting in a scalar (a single number). This operation involves multiplying the corresponding components of the vectors and then summing those products. For vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), the dot product is calculated as:
\[\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3\]In our problem, by choosing the vector \( \mathbf{v} = \langle 1, 1, 1 \rangle \), the dot product becomes a simple sum of the components of \( \mathbf{u} \), specifically \( u_1 + u_2 + u_3 \). The choice of \( \mathbf{v} \) plays a pivotal role in manipulating the inequality, as we can see here.
The dot product not only helps in computational optimizations, but also offers a geometric insight, as it reflects the cosine of the angle between two vectors when normalized. This is encapsulated in the relation: \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos \theta \).
This property allows the dot product to be a foundational concept in understanding vector projections and angles between vectors in vector algebra.

Vector Magnitude

The magnitude of a vector is a measure of its length in space, often referred to as its "norm" or "length." It gives us an idea of how "big" the vector is irrespective of its direction. For a vector \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \), its magnitude \( |\mathbf{u}| \) is calculated using the Euclidean formula:
\[|\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2}\]Understanding vector magnitude is essential in both physics and mathematics because it helps quantify an object's scale in multi-dimensional spaces.
In the exercise, calculating the magnitude of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is vital for applying the Cauchy-Schwarz Inequality. When we find \( |\mathbf{v}| \), with \( \mathbf{v} = \langle 1, 1, 1 \rangle \), it results in \( \sqrt{3} \), which emphasizes how this choice generates a standard scalar multiplier in inequalities.
This magnitude component from both vectors lays the groundwork for comparing the size of products, showing how the inequality remains valid by comparing the squared magnitude of their sum with the product's terms.

Vector Algebra

Vector algebra encompasses the operations involving vectors, including addition, scalar multiplication, and the dot product, among others. These operations are incredibly helpful when solving problems that involve forces, motion, and other physical phenomena.
Vectors can be added by summing their corresponding components; multiplicative operations scale vectors or project them in certain directions. Operations are guided by rules that mirror the properties of numbers, such as the distributive law.
In the context of our exercise, vector algebra allows us to manipulate vectors \( \mathbf{u} \) and \( \mathbf{v} \) under the Cauchy-Schwarz Inequality to derive useful results from the components. When the dot product is combined with vector magnitudes within the Cauchy-Schwarz framework, it reveals the relationship between overall vector alignment and their individual magnitudes.
Given vectors can be adapted creatively (as in choosing \( \mathbf{v} = \langle 1, 1, 1 \rangle \)) to explore expressions and compare magnitudes, vector algebra becomes a powerful tool to unlock complex insights. This is why systematically using these principles can efficiently solve real-world problems, from engineering to computer graphics.
Ultimately, a lot of physics and computer-related tasks are fundamentally about adjusting vectors for optimal alignment or motion, demonstrating vector algebra's foundational role.