Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot \mathbf{u}=\|\mathbf{u}\|^{2}\)

Step-by-Step Solution

Answer

\( \mathbf{u} \cdot \mathbf{u} = \| \mathbf{u} \|^2 \) is proved using the definition of dot product and magnitude.

Step 1: Understand the Dot Product

The dot product of two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) is defined as the sum of the products of their corresponding components: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \). In this property, we are interested in calculating \( \mathbf{u} \cdot \mathbf{u} \).

Step 2: Calculate the Dot Product of \( \mathbf{u} \) with Itself

Using the definition from Step 1, substitute \( \mathbf{u} \) for both vectors in the dot product: \( \mathbf{u} \cdot \mathbf{u} = (u_1 \cdot u_1) + (u_2 \cdot u_2) = u_1^2 + u_2^2 \).

Step 3: Recall the Magnitude of a Vector

The magnitude (or norm) of a vector \( \mathbf{u} = \langle u_1, u_2 \rangle \) is given by \( \| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2} \). The square of the magnitude is \( \| \mathbf{u} \|^2 = u_1^2 + u_2^2 \).

Step 4: Compare the Results

From Step 2, we have \( \mathbf{u} \cdot \mathbf{u} = u_1^2 + u_2^2 \) and from Step 3, \( \| \mathbf{u} \|^2 = u_1^2 + u_2^2 \). Therefore, these two expressions are equal, proving that \( \mathbf{u} \cdot \mathbf{u} = \| \mathbf{u} \|^2 \).

Key Concepts

Two-dimensional Vectors

Two-dimensional vectors are essential in mathematics and physics as they allow us to describe quantities with both magnitude and direction in a plane. Think of them as arrows in a two-dimensional space, defined by their endpoints.
These vectors are often represented using coordinates, like \((x, y)\).

• A vector has two components that describe its position relative to the origin.
• The components can be denoted in various forms, such as \([u_1, u_2]\) or \((u_1, u_2)\).

Understanding two-dimensional vectors is crucial for navigating problems that involve direction and magnitude, such as calculating force or velocity in physics.

Vector Magnitude

The magnitude of a vector, often known as its length or size, tells us how long the vector is. For a vector \(\mathbf{u} = \langle u_1, u_2 \rangle\), the magnitude is calculated using the formula:

• \(\| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2} \)

This expression is derived from the Pythagorean theorem applied to the right triangle formed by the vector and its components.
The magnitude essentially gives the Euclidean distance from the vector's tail to its tip in the Cartesian plane. Knowing a vector's magnitude is essential in determining the vector's strength or speed if it symbolizes physical quantities.

Vector Norm

The term vector norm is often interchangeable with vector magnitude, particularly when discussing the standard or "Euclidean" norm. In the context of a vector \( \mathbf{u} = \langle u_1, u_2 \rangle \), the norm is expressed as:

• \(\| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2} \)

This concept extends beyond just measuring length; it can also apply to any linear combination of dimensions, not just two-dimensional space.
Norms provide a consistent way to measure vectors' magnitude in different contexts, like multi-dimensional spaces or in functional analysis. Understanding the norm of a vector is crucial for determining metrics like distance and size in various settings.

Vector Components

Vector components allow for the decomposition of vectors into simpler parts, enabling easier calculation and application in various problems. For any vector \((\mathbf{u})\) represented as \(\langle u_1, u_2 \rangle\), its components can tell us how the vector relates to its axes of reference.

• \(u_1\) is the projection of the vector on the x-axis.
• \(u_2\) is the projection of the vector on the y-axis.

These components are fundamental in reconstructing a vector from its influences in each axis direction. Components simplify problems by breaking vectors into manageable parts that can be independently analyzed. This decomposition is key in applying mathematical operations such as the dot product, which involves multiplying corresponding components of two vectors.