Problem 30 - University Calculus: Early Transcendentals

Question

Cancelation 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, you cannot conclude \( \mathbf{v}_1 = \mathbf{v}_2 \).

Step 1: Understanding the Dot Product

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \). If the vectors are not zero, the dot product results in a scalar and is commutative (i.e., \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \)).

Step 2: Analyzing Given Equation with Dot Product

We need to compare the given condition \( \mathbf{u} \cdot \mathbf{v}_1 = \mathbf{u} \cdot \mathbf{v}_2 \) to standard multiplication. In normal multiplication, if \( u \cdot v_1 = u \cdot v_2 \) and \( u eq 0 \), we can conclude \( v_1 = v_2 \). However, dot product involves the magnitudes and direction of vectors, not just real number multiplication.

Step 3: Counterexample with Orthogonal Components

Consider \( \mathbf{u} = \begin{bmatrix} 1 \ 0 \end{bmatrix} \), \( \mathbf{v}_1 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \), and \( \mathbf{v}_2 = \begin{bmatrix} 1 \ -1 \end{bmatrix} \). Both have \( \mathbf{u} \cdot \mathbf{v}_1 = 1 \) and \( \mathbf{u} \cdot \mathbf{v}_2 = 1 \), since the second components of \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) contribute 0 due to the zero in \( \mathbf{u} \). Yet, clearly \( \mathbf{v}_1 eq \mathbf{v}_2 \).

Step 4: Conclusion on Cancellation in Dot Products

The counterexample shows that even if \( \mathbf{u} \cdot \mathbf{v}_1 = \mathbf{u} \cdot \mathbf{v}_2 \) with \( \mathbf{u} eq \mathbf{0} \), it is not necessarily true that \( \mathbf{v}_1 = \mathbf{v}_2 \). Therefore, the same cancellation rule in real-number multiplication does not apply to dot products.

Key Concepts

Vector Multiplication

Understanding vector multiplication is crucial for mastering vector algebra. Unlike simple scalar multiplication, where numbers are multiplied, vector multiplication involves directions and magnitudes, giving it unique properties. In vector addition, components are added separately to create a new vector. However, vector multiplication can occur in a few forms, most notably the dot product (or scalar product) and the cross product.
The dot product, the focus of this discussion, involves multiplying corresponding components of two vectors and summing the results. This operation is foundational in vector algebra as it provides a way to quantify how much two vectors align with each other. The formula for the dot product of vectors \( \mathbf{a} = (a_1, a_2, \ldots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \ldots, b_n) \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \)

This result yields a scalar, explaining why it's often called the scalar product. A positive result indicates that vectors point in a similar direction, while a result of zero suggests orthogonality, or perpendicular vectors.
Understanding these calculations is vital to solving more complex problems involving vectors in various fields such as physics, computer graphics, and more.

Orthogonal Vectors

Orthogonality is an essential concept in vector mathematics. Two vectors are considered orthogonal if their dot product is zero. This means they are perpendicular to each other at a 90-degree angle. For example, in a two-dimensional space, the vectors \(\mathbf{a} = (1, 0)\) and \(\mathbf{b} = (0, 1)\) are orthogonal since their dot product is:

• \( \mathbf{a} \cdot \mathbf{b} = (1 \cdot 0) + (0 \cdot 1) = 0 \)

Orthogonality signifies that the vectors do not influence each other in terms of projection along one another. This concept is fundamental, especially in spaces involving multiple dimensions, where orthogonality may imply physical independence, as in the case of forces, or independence in data sets.
Understanding orthogonal vectors is crucial in contexts like computer graphics for normalizing surfaces or in physics for resolving force vectors. Mastering this concept is key for anyone working with vector spaces and is invaluable across STEM fields.

Scalar Product

The scalar product, more commonly known as the dot product, is a unique type of vector multiplication that results in a scalar rather than another vector. This operation is mathematical shorthand for projecting one vector onto another. It combines both magnitudes and the cosine of the angle between the vectors. The scalar product is defined as:

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

where \( \theta \) is the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \), and \( \|\mathbf{a}\| \) and \( \|\mathbf{b}\| \) are their magnitudes. The result tells us how much of one vector is "in the direction" of another.
When the scalar product equals zero, the vectors are orthogonal. This property is not only essential in theoretical math but also in practical applications involving engineering, physics, and computer science. It's vital for students to grasp this concept to better understand problems involving vector projections, work done in physics, and other real-world applications.