Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that \(\mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}\).

Step-by-Step Solution

Answer

The given equation is proved using the expansion of squared norms and algebraic manipulation.

Step 1: Express Dot Product in Two-Dimensional Space

Let's consider the vectors \(\mathbf{u}\) and \(\mathbf{v}\) in a two-dimensional space where \(\mathbf{u} = [u_1, u_2]\) and \(\mathbf{v} = [v_1, v_2]\). Using the definition of the dot product, we have \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2\).

Step 2: Compute the Norms of Vector Additions

Calculate the norm of the vector \(\mathbf{u} + \mathbf{v}\) as: \[\|\mathbf{u} + \mathbf{v}\|^2 = \left((u_1 + v_1)^2 + (u_2 + v_2)^2\right).\] Similarly, find the norm of \(\mathbf{u} - \mathbf{v}\) as: \[\|\mathbf{u} - \mathbf{v}\|^2 = \left((u_1 - v_1)^2 + (u_2 - v_2)^2\right).\]

Step 3: Expand the Squared Norms

Expand \(\|\mathbf{u} + \mathbf{v}\|^2\): \[((u_1 + v_1)^2 + (u_2 + v_2)^2) = (u_1^2 + 2u_1v_1 + v_1^2) + (u_2^2 + 2u_2v_2 + v_2^2)\]. Similarly, expand \(\|\mathbf{u} - \mathbf{v}\|^2\): \[(u_1 - v_1)^2 + (u_2 - v_2)^2 = (u_1^2 - 2u_1v_1 + v_1^2) + (u_2^2 - 2u_2v_2 + v_2^2)\].

Step 4: Perform Algebraic Manipulation to Isolate Dot Product

Subtract the expansions obtained in Step 3: \[((u_1^2 + 2u_1v_1 + v_1^2) + (u_2^2 + 2u_2v_2 + v_2^2)) - ((u_1^2 - 2u_1v_1 + v_1^2) + (u_2^2 - 2u_2v_2 + v_2^2))\]which simplifies to \[4(u_1v_1 + u_2v_2)\].Thus:\[\frac{1}{4}\|\mathbf{u} + \mathbf{v}\|^2 - \frac{1}{4}\|\mathbf{u} - \mathbf{v}\|^2 = u_1v_1 + u_2v_2 = \mathbf{u} \cdot \mathbf{v}.\]

Step 5: Conclude the Proof

Since we have found \[u_1v_1 + u_2v_2 = \mathbf{u} \cdot \mathbf{v}\] from our calculations, we have successfully shown that \[\mathbf{u} \cdot \mathbf{v} = \frac{1}{4}\|\mathbf{u} + \mathbf{v}\|^2 - \frac{1}{4}\|\mathbf{u} - \mathbf{v}\|^2.\] This proves the given statement.

Key Concepts

Vector Norm

In mathematics, a vector norm is a fundamental concept that helps us measure a vector's length or magnitude. Consider a vector represented as \( \mathbf{u} = [u_1, u_2] \) in a two-dimensional space. The norm of this vector, often denoted by \( \|\mathbf{u}\| \), is calculated using the formula:\[ \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2} \]When dealing with problems that involve the addition or subtraction of vectors, such as the vector sum \( \mathbf{u} + \mathbf{v} \), we can find the norm of the resulting vector by computing its magnitude:\[ \|\mathbf{u} + \mathbf{v}\|^2 = (u_1 + v_1)^2 + (u_2 + v_2)^2 \]Similarly, for the vector difference \( \mathbf{u} - \mathbf{v} \), the squared norm is:\[ \|\mathbf{u} - \mathbf{v}\|^2 = (u_1 - v_1)^2 + (u_2 - v_2)^2 \]Breaking down these formulas helps us better understand the properties and characteristics of vector operations.

Algebraic Manipulation

Algebraic manipulation is a powerful technique used in mathematics to rearrange and simplify expressions. In this exercise, we utilize algebraic manipulation to prove the given statement.
It involves breaking down expressions into simpler components or rewriting equations in a form that reveals hidden insights. Let's look at how it applies to this problem:

• Start by expanding the norms: \( \|\mathbf{u} + \mathbf{v}\|^2 \) and \( \|\mathbf{u} - \mathbf{v}\|^2 \).
• In our case, the expansion results in: \( (u_1^2 + 2u_1v_1 + v_1^2) + (u_2^2 + 2u_2v_2 + v_2^2) \) for \( \|\mathbf{u} + \mathbf{v}\|^2 \).
• For \( \|\mathbf{u} - \mathbf{v}\|^2 \), it expands to: \( (u_1^2 - 2u_1v_1 + v_1^2) + (u_2^2 - 2u_2v_2 + v_2^2) \).

Now, subtract the second expanded norm from the first to eliminate terms that do not involve products of components of \( \mathbf{u} \) and \( \mathbf{v} \). Performing these steps carefully, you will reveal that the result simplifies to the dot product of \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \), which proves the exercise statement.

Two-Dimensional Vectors

Two-dimensional vectors are an essential part of vector algebra, especially when analyzing geometric problems or performing operations in algebraic proofs. Vectors in two-dimensional space are typically represented as \( \mathbf{u} = [u_1, u_2] \) or \( \mathbf{v} = [v_1, v_2] \), where each component corresponds to an axis in a plane.

• When dealing with 2D vectors, we can perform various operations such as addition, subtraction, and calculate the dot product.

For example, the dot product in 2D is a straightforward multiplication and sum of the components:\[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \]In geometric terms, the dot product helps in evaluating the angle between vectors, determining their orthogonality, or calculating projections.
Understanding the behavior of two-dimensional vectors is crucial, as it simplifies visualization and computation in various mathematical and physical contexts, making it easier to grasp more complex vector operations.