Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Given the two nonparallel vectors \(\mathbf{a}=-4 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{b}=2 \mathbf{i}-\mathbf{j}\) and another vector \(\mathbf{r}=6 \mathbf{i}-7 \mathbf{j}\), find scalars \(k\) and \(m\) such that \(\mathbf{r}=k \mathbf{a}+m \mathbf{b}\).

Step-by-Step Solution

Answer

The scalars are \( k = -4 \) and \( m = -5 \).

Step 1: Represent the equation

Write the given condition as a vector equation. We have: \( \mathbf{r} = k \mathbf{a} + m \mathbf{b} \), which translates to \( (6 \mathbf{i} - 7 \mathbf{j}) = k(-4 \mathbf{i} + 3 \mathbf{j}) + m(2 \mathbf{i} - \mathbf{j}) \).

Step 2: Break down into components

Break the equation into components: equate the \(\mathbf{i}\) components and \(\mathbf{j}\) components separately. The equation becomes: \( 6 = -4k + 2m \) (for \(\mathbf{i}\)) and \( -7 = 3k - m \) (for \(\mathbf{j}\)).

Step 3: Solve the first equation for one variable

From the equation \( 6 = -4k + 2m \), solve for one of the variables. Solve for \(m\): \( 2m = 6 + 4k \) leading to \( m = 3 + 2k \).

Step 4: Substitute and solve the second equation

Substitute \( m = 3 + 2k \) into the second equation \( -7 = 3k - m \). This becomes \( -7 = 3k - (3 + 2k) \) which simplifies to \( -7 = k - 3 \).

Step 5: Solve for k

Solve \( -7 = k - 3 \) to find \( k = -4 \).

Step 6: Find m using k

Substitute \( k = -4 \) into \( m = 3 + 2k \), yielding \( m = 3 + 2(-4) = 3 - 8 = -5 \).

Step 7: Verify the solution

Verify that the values of \( k \) and \( m \) satisfy both vector component equations: \( 6 = -4(-4) + 2(-5) = 16 - 10 = 6 \) and \( -7 = 3(-4) - (-5) = -12 + 5 = -7 \). The solution is verified.

Key Concepts

Vector Components

Vectors are fundamental in understanding many physical phenomena. A vector represents both a magnitude and a direction. Often, a vector is broken down into simpler parts called components. This makes it easier to handle mathematically.

Consider a vector described in a two-dimensional plane by components along the axes. In our exercise, vectors are expressed as combinations of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which point in the x and y directions respectively.

• \( \mathbf{a} = -4 \mathbf{i} + 3 \mathbf{j} \): Here, the components are \(-4\) in the x-direction and \(3\) in the y-direction.
• \( \mathbf{b} = 2 \mathbf{i} - \mathbf{j} \): This vector has components \(2\) in the x-direction and \(-1\) in the y-direction.
• \( \mathbf{r} = 6 \mathbf{i} - 7 \mathbf{j} \): \(6\) is the x-component and \(-7\) is the y-component.

By breaking vectors into components, we can simplify the problem to solving for scalars like \( k \) and \( m \), as in the given exercise. This decomposition leads to a system of linear equations that can be solved individually for each component direction.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is a real number. This operation stretches or shrinks the vector. For example, multiplying the vector \( \mathbf{a} = -4 \mathbf{i} + 3 \mathbf{j} \) by \( k \) gives a new vector \( k \mathbf{a} = k(-4 \mathbf{i} + 3 \mathbf{j}) = -4k \mathbf{i} + 3k \mathbf{j} \).

The result is:

• The vector's direction remains the same unless the scalar is negative, which reverses the direction.
• The vector's magnitude changes proportionally to the absolute value of the scalar.

In the exercise, scalar multiplication is necessary to express \( \mathbf{r} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \). The goal is to solve for the scalars \( k \) and \( m \) such that the combination matches \( \mathbf{r} \). This requires accurate manipulation and understanding of how scalars interact with vector components.

Linear Combination

A linear combination refers to an expression constructed from a set of terms by multiplying each term by a constant and adding the results. In vectors, this means combining vectors using scalar multiplication to find another vector.

For instance, our goal is to express the vector \( \mathbf{r} = 6 \mathbf{i} - 7 \mathbf{j} \) as a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \). Simply put, we want to find scalars \( k \) and \( m \) such that:

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

Separately equating the components of \( \mathbf{r} \) with those of \( k \mathbf{a} + m \mathbf{b} \) leads to a system of equations, namely \( 6 = -4k + 2m \) and \( -7 = 3k - m \).

Solving these equations determines the precise scalars needed to accurately represent \( \mathbf{r} \) using \( \mathbf{a} \) and \( \mathbf{b} \). This practice is crucial in various fields such as physics and engineering, where expressing complex systems as linear combinations simplifies analysis and problem-solving.