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: Write the equation for vector \( \mathbf{r} \)

We need to express \( \mathbf{r} = 6 \mathbf{i} - 7 \mathbf{j} \) as a linear combination of vectors \( \mathbf{a} = -4 \mathbf{i} + 3 \mathbf{j} \) and \( \mathbf{b} = 2 \mathbf{i} - \mathbf{j} \). This means we are looking for scalars \( k \) and \( m \) such that \( \mathbf{r} = k \mathbf{a} + m \mathbf{b} \). Thus, our equation is:\[ 6 \mathbf{i} - 7 \mathbf{j} = k(-4 \mathbf{i} + 3 \mathbf{j}) + m(2 \mathbf{i} - \mathbf{j}) \].

Step 2: Expand the right-hand side

Expand the terms on the right-hand side of the equation:\[ k(-4 \mathbf{i} + 3 \mathbf{j}) + m(2 \mathbf{i} - \mathbf{j}) = ( -4k + 2m ) \mathbf{i} + ( 3k - m ) \mathbf{j} \].

Step 3: Equate components

Now, equate the components of \( \mathbf{i} \) and \( \mathbf{j} \) from both sides of the equation:For \( \mathbf{i} \):\[ -4k + 2m = 6 \]For \( \mathbf{j} \):\[ 3k - m = -7 \].

Step 4: Solve the system of equations

Solve the system of equations:1. \[ -4k + 2m = 6 \]2. \[ 3k - m = -7 \]From the first equation, we can express \( m \) in terms of \( k \):\[ 2m = 6 + 4k \]\[ m = 3 + 2k \].Substitute \( m = 3 + 2k \) into the second equation:\[ 3k - (3 + 2k) = -7 \]Simplify:\[ 3k - 3 - 2k = -7 \]\[ k - 3 = -7 \]\[ k = -4 \].

Step 5: Find \( m \) using \( k \)

Substitute \( k = -4 \) back into \( m = 3 + 2k \):\[ m = 3 + 2(-4) \]\[ m = 3 - 8 \]\[ m = -5 \].

Step 6: Verify the solution

Substitute \( k = -4 \) and \( m = -5 \) back into the linear combination to check:\[ -4(-4) + 2(-5) = 16 - 10 = 6 \] (for \( \mathbf{i} \))\[ 3(-4) - (-5) = -12 + 5 = -7 \] (for \( \mathbf{j} \))Both values match \( \mathbf{r} = 6\mathbf{i} - 7\mathbf{j} \), confirming our solution is correct.

Key Concepts

Understanding Vector Equations

In the realm of linear algebra, vectors can be expressed through equations known as vector equations. These equations aim to represent one vector as a combination of other vectors. The idea is simple: you're trying to see if a certain vector, let's call it \( \mathbf{r} \), can be expressed as a sum of scaled versions of other vectors, like \( \mathbf{a} \) and \( \mathbf{b} \). In this example, \( \mathbf{r} = 6 \mathbf{i} - 7 \mathbf{j} \) is expressed as a linear combination of vectors \( \mathbf{a} = -4 \mathbf{i} + 3 \mathbf{j} \) and \( \mathbf{b} = 2 \mathbf{i} - \mathbf{j} \). The equation here is \( \mathbf{r} = k\mathbf{a} + m\mathbf{b} \), where \( k \) and \( m \) are the scalars we want to find.
In practice, such vector equations tell us that the given vector \( \mathbf{r} \) can be constructed using the available vectors \( \mathbf{a} \) and \( \mathbf{b} \) by scaling them appropriately, followed by summing them up. Finding the scalars is what solves the equation, showing the weights needed for each vector in the combination.

Linking to Systems of Linear Equations

To solve the vector equation \( \mathbf{r} = k\mathbf{a} + m\mathbf{b} \), you convert it into a system of linear equations. This is done by breaking down the vector equation into its components, leading to equations for each direction (\( \mathbf{i} \) and \( \mathbf{j} \)). Here, this results in two linear equations:

• \(-4k + 2m = 6\) for the \( \mathbf{i} \) component
• \(3k - m = -7\) for the \( \mathbf{j} \) component

These equations collectively form a system of linear equations. The system embodies the requirement that for the vectors combined on the right to equal \( \mathbf{r} \), both\( \mathbf{i} \) and \( \mathbf{j} \) components of the equation need to be satisfied. Solving this system helps in determining the exact values of \( k \) and \( m \).
Systems of linear equations are crucial because they provide a structured approach to finding intersections, alignments, or combinations in linear contexts. With the appropriate algebraic manipulations, you can find solutions that indicate whether a particular vector can be represented by a given set of vectors.

Solving for Scalars

Solving the system of equations derived from the vector equation means finding the values of the scalars \( k \) and \( m \). This involves substitution or elimination methods to simplify the expressions and isolate the variables.
Here, the equation for \( \mathbf{i} \) was solved first to express \( m \) in terms of \( k \), resulting in \( m = 3 + 2k \). This expression was then substituted into the \( \mathbf{j} \) equation. Simplifying this leads to finding \( k = -4 \). Once \( k \) is known, you can substitute back to find \( m = -5 \).

• Substitution Method: You solve one of the equations for one variable and substitute this into the other equation.
• Elimination Method: You eliminate one of the variables by adding or subtracting equations from each other.

These scalars, \( k = -4 \) and \( m = -5 \), provide the correct linear combination of \( \mathbf{a} \) and \( \mathbf{b} \) to obtain \( \mathbf{r} \). Verification is the final step; by substituting the scalars back into the original vector equation, confirming the left-hand side equals \( \mathbf{r} \). This verifies that the scalars found are indeed accurate.