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: Understanding Orthogonality Definition

Two vectors are orthogonal if their dot product equals zero. Given vectors are \( \mathbf{v_1} = 2c \mathbf{i} - 8 \mathbf{j} \) and \( \mathbf{v_2} = 3 \mathbf{i} + c \mathbf{j} \). We need to find \( c \) such that \( \mathbf{v_1} \cdot \mathbf{v_2} = 0 \).

Step 2: Calculate the Dot Product

The dot product formula for \( \mathbf{v_1} \) and \( \mathbf{v_2} \) is: \[ (2c \cdot 3) + (-8 \cdot c) = 6c - 8c. \]

Step 3: Set the Dot Product to Zero

Since \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are orthogonal, set the result of the dot product to zero: \[ 6c - 8c = 0. \]

Step 4: Solve for \( c \)

Simplifying the equation gives: \[ -2c = 0. \]Divide both sides by \(-2\) to solve for \( c \): \[ c = 0. \]

Step 5: Confirm the Solution

Plug \( c = 0 \) back into the vector components to ensure orthogonality. If \( c = 0 \), \( \mathbf{v_1} = 0 \mathbf{i} - 8 \mathbf{j} \) and \( \mathbf{v_2} = 3 \mathbf{i} + 0 \mathbf{j} \), and the dot product is indeed 0.

Key Concepts

Dot Product and Orthogonality

In the realm of vectors, the dot product is a key operation that allows us to measure various relationships between vectors.
Specifically, it helps determine if two vectors are orthogonal, meaning they form a right angle with each other. The dot product of two vectors, say \(\mathbf{v_1} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{v_2} = c\mathbf{i} + d\mathbf{j}\), is calculated using the formula:

• \(\mathbf{v_1} \cdot \mathbf{v_2} = ac + bd\)

This operation results in a scalar value. For vectors to be orthogonal, this scalar must equal zero. In our exercise, we were given the vectors \(2c \mathbf{i} - 8 \mathbf{j}\) and \(3 \mathbf{i} + c \mathbf{j}\). Calculating their dot product was essential to finding the condition for which they are orthogonal.

Understanding Vector Components

Vectors are often defined in terms of their components along the axes of a coordinate system. For instance, in a 2D plane, a vector can be expressed in terms of its \( \mathbf{i} \) (horizontal) and \( \mathbf{j} \) (vertical) components. Each component represents how much the vector stretches along the respective axis.Consider the vectors \(2c \mathbf{i} - 8 \mathbf{j}\) and \(3 \mathbf{i} + c \mathbf{j}\) from the exercise:

• The vector \(2c \mathbf{i} - 8 \mathbf{j}\) has a horizontal component \(2c\) and a vertical component \(-8\).
• The vector \(3 \mathbf{i} + c \mathbf{j}\) has a horizontal component \(3\) and a vertical component \(c\).

By analyzing these components, we can efficiently compute the dot product and understand how the vectors relate in space.

Solving Equations for Orthogonality

To find when two vectors are orthogonal, you set their dot product equal to zero and solve for the unknown variable. In our exercise, the given vectors were \(2c \mathbf{i} - 8 \mathbf{j}\) and \(3 \mathbf{i} + c \mathbf{j}\). The equation derived from their dot product was:

• \(6c - 8c = 0\)

Simplifying this, we combined like terms to get:

• \(-2c = 0\)

By dividing both sides by \(-2\), we found \(c = 0\).
Plugging \(c = 0\) back into the vector equations confirmed the vectors are orthogonal.
This step-by-step method illustrates how solving equations can help determine specific conditions for vector relationships such as orthogonality.