Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

For what numbers \(c\) and \(d\) are \(\mathbf{u}=c \mathbf{i}+\mathbf{j}+\mathbf{k}\) and \(\mathbf{v}=2 \mathbf{j}+d \mathbf{k}\) orthogonal?

Step-by-Step Solution

Answer

\(d = -2\) for any \(c\).

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is equal to zero. Hence, to find when the vectors \(\mathbf{u}\) and \(\mathbf{v}\) given by \(\mathbf{u} = c\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{v} = 2\mathbf{j} + d\mathbf{k}\) are orthogonal, we need to compute their dot product and set it equal to zero.

Step 2: Calculate the Dot Product

The dot product of \(\mathbf{u} = c\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{v} = 2\mathbf{j} + d\mathbf{k}\) is given by:\[(c\mathbf{i} + \mathbf{j} + \mathbf{k}) \cdot (2\mathbf{j} + d\mathbf{k}) = c \times 0 + 1 \times 2 + 1 \times d = 2 + d.\]

Step 3: Set the Dot Product to Zero

To find the values of \(c\) and \(d\) such that the vectors are orthogonal, set the result from Step 2 equal to zero:\[2 + d = 0\].

Step 4: Solve for \(d\)

Solve the equation \(2 + d = 0\) for \(d\):\[d = -2\].

Step 5: Verify the Role of \(c\)

Observe that the variable \(c\) does not appear in our equation. This indicates that the value of \(c\) does not affect the orthogonality condition. Hence, \(c\) can be any number.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental concept in vector mathematics. It provides a way to multiply two vectors, resulting in a scalar (a single number). The dot product is central in determining the relation between vectors, including understanding orthogonality. To compute the dot product of two vectors, each corresponding component is multiplied, and the results are summed up. For instance, for vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), the dot product is calculated as:\[ a_1b_1 + a_2b_2 + a_3b_3 \]In our specific scenario, with vectors \( \mathbf{u} = c\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \( \mathbf{v} = 2\mathbf{j} + d\mathbf{k}\), the computation involves matching components and observing that the components involving \( \mathbf{i} \) do not contribute (because there's no \( \mathbf{i} \)-component in \( \mathbf{v} \)).Make sure to remember that the result of the dot product provides insights into the angle between vectors. Particularly, when the result is zero, the vectors are orthogonal.

Vector Mathematics

Vectors are mathematical entities that have both a magnitude and a direction, making them incredibly useful for a wide range of applications, from physics to computer graphics. In In a Cartesian coordinate system, vectors are often expressed in terms of unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), which correspond to the x, y, and z axes, respectively. For any vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \), the values \( a_1, a_2, \) and \( a_3 \) represent its components in the respective direction.Vectors can be added together, subtracted, and multiplied in various ways, the dot product being one of them. Moreover, vectors are not limited to three dimensions; they can exist in any number of dimensions depending on the components they have. Nevertheless, each operation or mathematical manipulation follows similar principles regardless of the vector's dimension.Understanding and working with vectors involves not only manipulating their mathematical expressions but also visualizing their directions and magnitudes. This dual representation is what makes vector mathematics so powerfully versatile.

Orthogonality Condition

The orthogonality condition is a pivotal concept in vector mathematics. Two vectors are considered orthogonal if they are at a right angle to each other. Mathematically, this condition is achieved when their dot product is zero.This arises because the dot product geometrically correlates to multiplying the magnitudes of the vectors by the cosine of the angle between them. If the angle is 90 degrees, the cosine of this angle is zero, so the product is zero, confirming orthogonality.In our exercise, testing whether \( \mathbf{u} = c\mathbf{i} + \mathbf{j} + \mathbf{k} \) and \( \mathbf{v} = 2\mathbf{j} + d\mathbf{k} \) are orthogonal involves setting their dot product to zero. As calculated, \( 2 + d = 0 \) must be satisfied. Here, resolving \( d = -2 \) reinforces their orthogonality with no dependency on the component \( c \).Orthogonality is foundational in many applications, including solving systems of linear equations, projections, and even in fields like machine learning where orthogonal vectors represent uncorrelated features.