Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

Given the two nonparallel vectors \(\mathbf{a}=3 \mathbf{i}-2 \mathbf{j}\) and \(\mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\) and another vector \(\mathbf{r}=7 \mathbf{i}-8 \mathbf{j}\), find scalars \(k\) and \(m\) such that \(\mathbf{r}=k \mathbf{a}+m \mathbf{b}\).

Step-by-Step Solution

Answer

k = 3, m = -1.

Step 1: Express the Equation

We have the vector equation \( \mathbf{r} = k \mathbf{a} + m \mathbf{b} \). Substitute \( \mathbf{a} = 3\mathbf{i} - 2\mathbf{j} \), \( \mathbf{b} = -3\mathbf{i} + 4\mathbf{j} \) and \( \mathbf{r} = 7\mathbf{i} - 8\mathbf{j} \) into this equation to get: \[7\mathbf{i} - 8\mathbf{j} = k(3\mathbf{i} - 2\mathbf{j}) + m(-3\mathbf{i} + 4\mathbf{j})\] which can be expanded to \((3k - 3m)\mathbf{i} + (-2k + 4m)\mathbf{j}\).

Step 2: Set Up the System of Equations

From the equation \((3k - 3m)\mathbf{i} + (-2k + 4m)\mathbf{j} = 7\mathbf{i} - 8\mathbf{j}\), we derive the system of linear equations by comparing coefficients: \[3k - 3m = 7\] and \[-2k + 4m = -8\].

Step 3: Solve the First Equation for k

Take the equation \(3k - 3m = 7\). Divide the entire equation by 3 to simplify, yielding \(k - m = \frac{7}{3}\). Now express \(k\) in terms of \(m\): \[k = m + \frac{7}{3}\].

Step 4: Substitute and Solve for m

Substitute the expression for \(k\) from Step 3 into the second equation \(-2k + 4m = -8\): \[-2(m + \frac{7}{3}) + 4m = -8\]. Expand and simplify to find \(-2m - \frac{14}{3} + 4m = -8\). Simplify further: \ 2m - \frac{14}{3} = -8 \ 2m = -8 + \frac{14}{3} \ 2m = -\frac{24}{3} + \frac{14}{3} \ 2m = -\frac{10}{3} \ \therefore m = -\frac{5}{3}. "},{

Key Concepts

System of Linear Equations

In mathematics, a system of linear equations is a collection of two or more linear equations involving the same set of variables. In the context of our problem involving vectors, we have two equations derived from the given vector equation. These equations are created by comparing coefficients of each unit vector, such as \( \mathbf{i} \) and \( \mathbf{j} \). Here, our system is:

• \( 3k - 3m = 7 \)
• \( -2k + 4m = -8 \)

This system represents the relationships between the scalars \( k \) and \( m \). Solving this system will help us find values for \( k \) and \( m \) that allow vector \( \mathbf{r} \) to be expressed as a linear combination of vectors \( \mathbf{a} \) and \( \mathbf{b} \). Understanding how to set up and solve these equations is crucial for tackling real-world problems in fields like physics and engineering, where similar problems often arise.

Vector Algebra

Vector algebra is the field of mathematics concerning the algebraic operations involving vectors. Important operations include addition, subtraction, and scalar multiplication of vectors. In the given exercise, vector algebra helps us set up the linear combination that forms \( \mathbf{r} \) using vectors \( \mathbf{a} \) and \( \mathbf{b} \).

We express vector \( \mathbf{r} \) as a sum of scaled vectors:\[ \mathbf{r} = k \mathbf{a} + m \mathbf{b} \]Through substitution and expansion, we obtain the coefficients of each component:

• \( (3k - 3m) \mathbf{i} \)
• \( (-2k + 4m) \mathbf{j} \)

Vector algebra simplifies complex geometries in physical space, especially when dealing with forces or motion, for which expressing vectors as combinations is a fundamental skill.

Solving Equations

Solving equations involves finding values for variables that satisfy given mathematical statements. In our exercise, solving equations is a critical step to determine the values of \( k \) and \( m \).

We begin by simplifying the first equation, \( 3k - 3m = 7 \), by dividing each term by 3 to express \( k \) in terms of \( m \):\[ k = m + \frac{7}{3} \]This expression is substituted into the second equation, \( -2k + 4m = -8 \), replacing \( k \) and simplifying:

• Reorganize to find \( m \)
• Solve for \( m \) and substitute back to find \( k \)

Simplifying and solving such equations require careful manipulation and understanding of algebraic principles, serving as foundational techniques in mathematics necessary for computational solutions in various disciplines.