Question

Prove the Cauchy-Schwarz Inequality for two-dimensional vectors: $$ |\mathbf{u} \cdot \mathbf{v}| \leq\|\mathbf{u}\|\|\mathbf{v}\| $$

Short answer

The Cauchy-Schwarz inequality is proven by comparing the squared forms and taking square roots.

Step-by-step solution

Define the Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality for two-dimensional vectors \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \) is written as:\[|\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\|\|\mathbf{v}\|.\]Here, the dot product \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \) and the magnitudes are \( \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2} \) and \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2} \).

Express Dot Product and Magnitudes

Calculate the dot product:\[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2.\]Calculate the magnitudes:\[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}, \quad \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}.\]

Expand the Square of the Dot Product

Start with the expression \((\mathbf{u} \cdot \mathbf{v})^2\) and write it as:\[(u_1v_1 + u_2v_2)^2 = u_1^2v_1^2 + 2u_1v_1u_2v_2 + u_2^2v_2^2.\]

Use Magnitude Squared Comparison

We know:\[\|\mathbf{u}\|^2 \cdot \|\mathbf{v}\|^2 = (u_1^2 + u_2^2)(v_1^2 + v_2^2).\]Expanding this multiplication gives:\[u_1^2v_1^2 + u_1^2v_2^2 + u_2^2v_1^2 + u_2^2v_2^2.\]

Apply the Inequality Relationship

Notice the expression \((u_1v_1 + u_2v_2)^2\) we expanded in Step 3 can be related to \(\|\mathbf{u}\|^2 \cdot \|\mathbf{v}\|^2\) from Step 4 using:\[(u_1^2v_1^2 + 2u_1v_1u_2v_2 + u_2^2v_2^2) \leq (u_1^2v_1^2 + u_1^2v_2^2 + u_2^2v_1^2 + u_2^2v_2^2).\]The inequality above holds due to the non-negativity of squares and cross terms' cancelling property.

Conclude with Cauchy-Schwarz Inequality

Since the squared magnitudes and squares of dot products relate through inequality, we have:\[(u_1v_1 + u_2v_2)^2 \leq (u_1^2 + u_2^2)(v_1^2 + v_2^2).\]Taking the square root on both sides results in:\[|u_1v_1 + u_2v_2| \leq \sqrt{u_1^2 + u_2^2} \cdot \sqrt{v_1^2 + v_2^2}\]which gives us the Cauchy-Schwarz inequality:\[\|\mathbf{u} \cdot \mathbf{v}\| \leq \|\mathbf{u}\| \|\mathbf{v}\|.\]

Key concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector calculus. For two vectors, say \(\mathbf{u} = (u_1, u_2)\) and \(\mathbf{v} = (v_1, v_2)\), the dot product is calculated as \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2\). This product results in a scalar, not a vector.

The essence of the dot product is found in measuring how much one vector extends in the direction of another. It is crucial in various mathematical applications, including projections and determining angles between vectors.

• Zero dot product implies the vectors are perpendicular.
• Positive dot product indicates vectors are oriented similarly.
• Negative dot product suggests vectors have opposite directions.

Understanding this allows one to appreciate the geometric significance of vector operations, setting a foundation for more complex topics like the Cauchy-Schwarz Inequality.

Vector Magnitude

The magnitude, or length, of a vector represents how long the vector is in the coordinate space. Calculating it is simple yet significant, especially in physics and engineering. For a two-dimensional vector \(\mathbf{u} = (u_1, u_2)\), the magnitude is found through the equation \(\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}\).

Knowing a vector's magnitude involves using the Pythagorean theorem, where the components \(u_1\) and \(u_2\) are akin to the sides of a right triangle, and the magnitude corresponds to the hypotenuse.

• Vector magnitude gives a quantitative value for distance or length in spatial terms.
• It plays a critical role in normalizing vectors, which are vectors whose length is scaled to one.
• This concept helps measure and compare the relative sizes of vectors.

Magnitude is pivotal when applying the Cauchy-Schwarz Inequality as it determines vector norms or sizes.

Inequality Proof

Inequality proofs, like that of the Cauchy-Schwarz Inequality, illustrate the interplay between vectors in more abstract problems. This particular inequality ensures that the absolute value of the dot product of two vectors is never greater than the product of their magnitudes.

When proving this inequality, we often deal with algebraic expansions and simplifications. For example, expanding the squared dot product \((u_1v_1 + u_2v_2)^2\) reveals certain sums which must be carefully related to the magnitudes' product \((u_1^2 + u_2^2)\times(v_1^2 + v_2^2)\).

• Proofs involve breaking down complex expressions into simpler components.
• They require recognizing equivalent forms and using known identities and properties.
• Understanding each step clearly is crucial for correct derivation.

Inequality proofs offer a rigorous approach to confirming relationships between mathematical entities ensuring knowledge of vector constraints and properties.

Two-Dimensional Vectors

Two-dimensional vectors are fundamental components of linear algebra and geometry, lying in a plane with both an x and a y component, expressed as \(\mathbf{u} = (u_1, u_2)\).

These vectors help model and analyze real-world situations as they provide a straightforward representation of different quantities like force, velocity, and displacement in physics. Their simplicity does not undermine their utility across several applications.

• Two-dimensional vectors operate within the Cartesian coordinate system for geometrical interpretation.
• They help solve equations involving lines, shapes, motion, and forces.
• Understanding their basics is crucial before delving into multi-dimensional vector spaces.

These vectors are central in studying inequalities like the Cauchy-Schwarz Inequality, providing clarity on how algebraic relationships translate into geometric insights.