Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

For what numbers \(c\) are \(2 c \mathbf{i}-8 \mathbf{j}\) and \(3 \mathbf{i}+c \mathbf{j}\) orthogonal?

Step-by-Step Solution

Answer

The number \( c \) is 0.

Step 1: Recall the condition for orthogonality

Two vectors are orthogonal if their dot product is zero. The dot product of vectors \( a\mathbf{i} + b\mathbf{j} \) and \( c\mathbf{i} + d\mathbf{j} \) is calculated as \( ac + bd \).

Step 2: Apply the condition to the given vectors

Let the vectors be \( \mathbf{v}_1 = 2c\mathbf{i} - 8\mathbf{j} \) and \( \mathbf{v}_2 = 3\mathbf{i} + c\mathbf{j} \). Their dot product is \( (2c) \cdot (3) + (-8) \cdot (c) \).

Step 3: Set up the equation from the dot product

The dot product is \( 6c - 8c \). For the vectors to be orthogonal, this must be equal to zero. So, \( 6c - 8c = 0 \).

Step 4: Solve the equation

Simplify the equation to \( -2c = 0 \). Solving for \( c \), we divide both sides by \(-2\), which gives \( c = 0 \).

Key Concepts

Dot Product

In vector algebra, the dot product is a fundamental operation that combines two vectors to produce a scalar, or numerical, value. It is represented as the sum of the products of their corresponding components. For example, given two vectors,

• \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \)
• \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \)

The dot product is calculated as:

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

The result of a dot product tells us about the angle between the two vectors. If the dot product equals zero, it indicates that the vectors are orthogonal, meaning they are at a 90-degree angle to each other. This property is often used in physics and engineering to determine the perpendicularity of forces or directions.

Vector Algebra

Vector algebra deals with mathematical operations involving vectors. Vectors are quantities that have both magnitude and direction, different from scalars, which only possess magnitude. In vector algebra, we study:

• Addition: Combining two vectors to get a resultant vector by adding their components.
• Subtraction: Deducting one vector from another, similar to addition but with subtraction in position.
• Scalar multiplication: Multiplying a vector by a scalar adjusts its magnitude while maintaining the direction.

These operations are crucial for simplifying complex problems in physics and engineering. Vector algebra provides the tools to work with vectors in various applications, such as calculating resultant forces or defining displacements. Understanding vector operations is essential for fields like computer graphics and navigation systems.

Orthogonality Condition

Orthogonality is a key concept in vector algebra, most notably recognized by the condition that two vectors are orthogonal if their dot product is zero. This signifies that the vectors intersect at a perpendicular angle.

• For vectors \( \mathbf{v}_1 = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v}_2 = c\mathbf{i} + d\mathbf{j} \), the orthogonality condition is expressed as \( ac + bd = 0 \).

Orthogonality has practical applications in various fields including:

• Computer Science: In algorithms for data separation and clustering.
• Engineering: In designing and analyzing structure forces.

When vectors are orthogonal, they hold a unique independence from each other, making it easier to break down complex problems into simpler components. Recognizing orthogonality is crucial in optimizing operations and understanding spatial relationships.

Linear Algebra

Linear algebra is the branch of mathematics that studies vectors, vector spaces, and linear mappings. It forms the foundation for vector operations and discusses concepts like matrix manipulation and systems of linear equations.

• The vector space is a collection of vectors that can be scaled and added.
• Basis vectors: Set the axes in a vector space, defining directions and dimensions.
• Linear transformations: Functions that map vectors into another vector within the same space.

Linear algebra is essential in developing algorithms in computer science, solving physical equations in engineering, and managing multidimensional data in analytics. It gives us the framework to handle data efficiently and effectively, revealing insights through operations like those involving matrices and eigenvalues.