Question

Prove that the triangle inequality \(\|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\|\) holds for all vectors \(\mathbf{u}\) and \(\mathbf{v}\). [Hint: Consider the triangle with \(\mathbf{u}\) and \(\mathbf{v}\) as two sides. \(]\)

Short answer

The triangle inequality is proved using the Cauchy-Schwarz inequality and properties of norms.

Step-by-step solution

Understand the Triangle Inequality

The triangle inequality for vectors states that the length of any side of a triangle is less than or equal to the sum of the lengths of the other two sides. For vectors, this states that for any vectors \(\mathbf{u}\) and \(\mathbf{v}\), the norm (or length) of their sum is not greater than the sum of their norms.

Express the Norms

The norms of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are expressed as \(\|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}}\) and \(\|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}}\), where \(\cdot\) denotes the dot product.

Consider the Norm of the Sum

The norm of the sum \(\mathbf{u} + \mathbf{v}\) is given by \(\|\mathbf{u} + \mathbf{v}\| = \sqrt{(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v})}\). Expanding this expression using the distributive property of the dot product, we have \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v}) = \mathbf{u} \cdot \mathbf{u} + 2\mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{v}.\)

Apply Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that for any two vectors \(\mathbf{a}\) and \(\mathbf{b}\), \((\mathbf{a} \cdot \mathbf{b})^2 \leq (\mathbf{a} \cdot \mathbf{a})(\mathbf{b} \cdot \mathbf{b})\). Applying this to \(\mathbf{u} \cdot \mathbf{v}\), we get \((\mathbf{u} \cdot \mathbf{v})^2 \leq \|\mathbf{u}\|^2 \|\mathbf{v}\|^2\).

Prove the Triangle Inequality

Using the expanded expression, \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v}) = \|\mathbf{u}\|^2 + 2\mathbf{u}\cdot \mathbf{v} + \|\mathbf{v}\|^2 \). By applying the Cauchy-Schwarz inequality, \(2\mathbf{u} \cdot \mathbf{v} \leq 2\|\mathbf{u}\| \|\mathbf{v}\|\). Thus, \(\|\mathbf{u} + \mathbf{v}\|^2 \leq (\|\mathbf{u}\| + \|\mathbf{v}\|)^2.\) Taking square roots on both sides, we obtain \(\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|.\)

Key concepts

Vector Norms

Vector norms essentially measure the length or size of a vector. Imagine vectors as arrows in space. The norm of a vector is analogous to the length of this arrow. There are various types of vector norms, but the most common is the Euclidean norm, which is equivalent to our intuitive idea of length. It is represented by \( \|\mathbf{u}\| \) and is calculated as \( \|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}} \). The norm is always non-negative and provides a straightforward way to compare the magnitudes of different vectors.
Understanding vector norms is crucial when working with the triangle inequality. The triangle inequality quantifies the intuitive idea that taking a direct path is always shorter or equal to taking a detour through another vector. As such, norms provide the tools to measure the sides of the 'triangle' in vector space.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation involving two vectors that results in a scalar (a single number). Mathematically, for two vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product is defined as \( \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta \), where \( \theta \) is the angle between the two vectors.
This operation does not only provide the magnitude part but also takes into account the direction of the vectors relative to each other. It is crucial for deriving vector expressions like the expanded form used in the triangle inequality, \((\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v})\). By breaking it down into \( \mathbf{u} \cdot \mathbf{u} + 2\mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{v} \), the dot product reveals how the lengths and directions of \( \mathbf{u} \) and \( \mathbf{v} \) interact.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a powerful tool in mathematics that involves the dot product. It provides a bound on the dot product of two vectors. The inequality states that for any vectors \( \mathbf{a} \) and \( \mathbf{b} \), \((\mathbf{a} \cdot \mathbf{b})^2 \leq (\mathbf{a} \cdot \mathbf{a})(\mathbf{b} \cdot \mathbf{b})\).
This inequality is instrumental in proving the triangle inequality. When applied to \( \mathbf{u} \) and \( \mathbf{v} \), it turns \( 2\mathbf{u} \cdot \mathbf{v} \leq 2\|\mathbf{u}\| \|\mathbf{v}\| \). By using this result, we can establish that \( \|\mathbf{u} + \mathbf{v}\|^2 \leq (\|\mathbf{u}\| + \|\mathbf{v}\|)^2 \), which leads to the final statement of the triangle inequality. Thus, the Cauchy-Schwarz inequality serves as a bridge between the dot product and norms in proving fundamental properties like the triangle inequality.

Vector Addition

Vector addition is the operation of adding two vectors together to produce a third vector. It follows the tip-to-tail principle, where the end of the first vector is the starting point for the second vector. Mathematically, the addition of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is written as \( \mathbf{u} + \mathbf{v} \).
This concept is intuitive if you picture vectors as arrows: adding them results in a new arrow pointing from the start of the first to the end of the second when arranged in order. In the context of the triangle inequality, vector addition reflects the creation of a triangle with \( \mathbf{u} \) and \( \mathbf{v} \) as the two sides, while \( \mathbf{u} + \mathbf{v} \) forms the third side. This also underscores the importance of understanding the sum \( \|\mathbf{u} + \mathbf{v}\| \) as it represents the "direct" path between the start and end points of adding these vectors.