Problem 28 - Thomas Calculus

Question

Cancellation in dot products In real-number multiplication, if \(u v_{1}=u v_{2}\) and \(u \neq 0,\) we can cancel the \(u\) and conclude that \(v_{1}=v_{2 .}\) Does the same rule hold for the dot product? That is, if \(\mathbf{u} \cdot \mathbf{v}_{1}=\mathbf{u} \cdot \mathbf{v}_{2}\) and \(\mathbf{u} \neq \mathbf{0},\) can you conclude that \(\mathbf{v}_{1}=\mathbf{v}_{2} ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

No, the cancellation rule does not hold for dot products; different vectors can have the same dot product with another vector.

Step 1: Understand the Dot Product

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \cdots + a_nb_n \), where each component of both vectors is multiplied and the products are summed up. It can also be expressed as \( \mathbf{a} \cdot \mathbf{b} = ||\mathbf{a}|| \ ||\mathbf{b}|| \ \cos \theta \), where \( \theta \) is the angle between the two vectors.

Step 2: Analyze the Given Equation

We are given that \( \mathbf{u} \cdot \mathbf{v}_{1} = \mathbf{u} \cdot \mathbf{v}_{2} \). Rewriting in terms of magnitudes and angles, we have: \( ||\mathbf{u}|| \ ||\mathbf{v}_1|| \cos \theta_1 = ||\mathbf{u}|| \ ||\mathbf{v}_2|| \cos \theta_2 \). Since \( \mathbf{u} eq \mathbf{0} \), it implies \( ||\mathbf{u}|| eq 0 \), allowing us to divide both sides by \( ||\mathbf{u}|| \), resulting in: \( ||\mathbf{v}_1|| \cos \theta_1 = ||\mathbf{v}_2|| \cos \theta_2 \).

Step 3: Consider Possibilities for \( \textbf{v}_1 \) and \( \textbf{v}_2 \)

From \( ||\mathbf{v}_1|| \cos \theta_1 = ||\mathbf{v}_2|| \cos \theta_2 \), it's clear that either the magnitudes or the angles relative to \( \mathbf{u} \) could be different for \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). Hence, different vectors can still satisfy this equation if their projections onto \( \mathbf{u} \) are the same.

Step 4: Conclude about Equality

Due to the existence of multiple angles that might satisfy \( ||\mathbf{v}_1|| \cos \theta_1 = ||\mathbf{v}_2|| \cos \theta_2 \), we cannot generally conclude that \( \mathbf{v}_1 = \mathbf{v}_2 \) solely from the given equality of dot products. Therefore, the cancellation property like multiplication of real numbers does not hold for the dot product.

Key Concepts

Vector Equality

Vector equality is a concept where two vectors are considered equal if they have identical magnitudes and directions. This implies that each corresponding component of the vectors must be equal. For vectors \( \mathbf{v}_1 = (v_{1x}, v_{1y}, v_{1z}) \) and \( \mathbf{v}_2 = (v_{2x}, v_{2y}, v_{2z}) \), if \( \mathbf{v}_1 = \mathbf{v}_2 \), then:

• Each component must be identical: \( v_{1x} = v_{2x}, v_{1y} = v_{2y}, v_{1z} = v_{2z} \)

In the dot product scenario discussed, we mistakenly assume that similar outcomes occur from the similarity in dot product values. However, just because two vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) result in the same dot product with a third vector \( \mathbf{u} \), it doesn't guarantee that

• Their individual components or directions are the same.
• The magnitudes and angles relative to \( \mathbf{u} \) might differ.

Therefore, vector equality requires a complete one-to-one correspondence of components and orientation, beyond mere equality of projection results as indicated by a dot product.

Angle Between Vectors

The angle between vectors is a measure of how two vectors point relative to each other. For any two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the angle \( \theta \) can be found using the dot product formula:

• \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| ||\mathbf{b}||} \)

This formula allows us to find the angle, provided that neither vector is the zero vector. The angle \( \theta \) gives insight into whether vectors are:

• Perpendicular if \( \theta = 90^\circ \)
• Parallel or anti-parallel if \( \theta = 0^\circ \) or \( 180^\circ \)

In the context of the exercise, even if \( \mathbf{u} \cdot \mathbf{v}_1 = \mathbf{u} \cdot \mathbf{v}_2 \), the angles \( \theta_1 \) and \( \theta_2 \) between \( \mathbf{u} \) and the respective \( \mathbf{v} \) vectors might differ altogether. This means similar dot products do not inherently imply the same directional relationships around the vector \( \mathbf{u} \). Thus, discrepancies in angles illustrate why dot product equality is not sufficient for vector equality.

Magnitude of Vectors

The magnitude or length of a vector describes its size, often representing how far from zero it stretches in space. For a vector \( \mathbf{v} = (v_x, v_y, v_z) \), its magnitude is given by:

• \( ||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2 + v_z^2} \)

Magnitude is fundamental in vector calculations, including the dot product, where it affects how vector quantities scale relative to each other. Returning to the exercise’s context, when you equate two dot products:

• Equality of magnitude does not assure equality of vector identity.
• \( ||\mathbf{v}_1|| \cos \theta_1 = ||\mathbf{v}_2|| \cos \theta_2 \) indicates potential variance in either magnitude or angular placement.

This relationship often denotes how vectors align with another indexing vector, \( \mathbf{u} \) in this case. Thus, equal dot products only confirm equivalent projections onto a particular line, but not necessarily matching magnitude across differing orientations.