Problem 99 - Precalculus

Question

A vector \(\mathbf{u}\) is a linear combination of \(\mathbf{p}\) and \(\mathbf{q}\) if there exist constants \(c_{1}\) and \(c_{2}\) such that \(\mathbf{u}=c_{1} \mathbf{p}+c_{2} \mathbf{q} .\) Show that \langle-6,4\rangle is a linear combination of \langle-8,4\rangle and \langle 1,-1\rangle

Step-by-Step Solution

Answer

Yes, \( \langle -6, 4 \rangle \) is a linear combination of \( \langle -8, 4 \rangle \) and \( \langle 1, -1 \rangle \) with \( c_1 = \frac{1}{2} \) and \( c_2 = -2 \).

Step 1: Set up the equation for linear combination

We want to express \( \langle -6, 4 \rangle \) as a linear combination of vectors \( \langle -8, 4 \rangle \) and \( \langle 1, -1 \rangle \). This means we need to find constants \( c_1 \) and \( c_2 \) such that: \[ \langle -6, 4 \rangle = c_1 \langle -8, 4 \rangle + c_2 \langle 1, -1 \rangle. \] In terms of components, this equation becomes: \[ (-6, 4) = c_1(-8, 4) + c_2(1, -1). \]

Step 2: Expand and equate the vector components

Expand the vectors on the right side of the equation: \[ c_1(-8, 4) + c_2(1, -1) = (-8c_1 + c_2, 4c_1 - c_2). \] Equate each component with the corresponding component of \( \langle -6, 4 \rangle \). This gives us the system of equations: \[ -8c_1 + c_2 = -6 \] \[ 4c_1 - c_2 = 4. \]

Step 3: Solve the system of equations

To solve the system of equations: \( -8c_1 + c_2 = -6 \) and \( 4c_1 - c_2 = 4 \), let's add the two equations together to eliminate \( c_2 \): \[ (-8c_1 + c_2) + (4c_1 - c_2) = -6 + 4. \] Simplifying gives: \[-4c_1 = -2. \] Solve for \( c_1 \): \[ c_1 = \frac{-2}{-4} = \frac{1}{2}. \]

Step 4: Substitute back to find the second constant

Substitute \( c_1 = \frac{1}{2} \) back into one of the original equations to find \( c_2 \). Using \( -8c_1 + c_2 = -6 \): \[ -8 \times \frac{1}{2} + c_2 = -6 \] \[ -4 + c_2 = -6 \] \[ c_2 = -6 + 4 = -2. \]

Step 5: Verify the solution

The solution for \( c_1 \) and \( c_2 \) is \( c_1 = \frac{1}{2} \) and \( c_2 = -2 \). Let's verify by substituting back into the equation for \( \langle -6, 4 \rangle \): \[ c_1 \langle -8, 4 \rangle + c_2 \langle 1, -1 \rangle = \frac{1}{2} \langle -8, 4 \rangle - 2 \langle 1, -1 \rangle. \] Calculate this: \[ \langle -4, 2 \rangle + \langle -2, 2 \rangle = \langle -6, 4 \rangle. \] This confirms our solution is correct.

Key Concepts

Vector Components

Vector components are essential in understanding the representation of vectors in a coordinate system. A vector in two dimensions can be expressed as a pair of numbers which denote its projections along the x and y axes. For instance, the vector \(\langle -6, 4\rangle\) indicates it has an x-component of -6 and a y-component of 4.

When dealing with vector components, it's crucial to understand that every vector can be broken down into horizontal and vertical parts. These parts tell us how far and in what direction the vector stretches along each axis.

In the context of a linear combination, we aim to express a vector via the sum of scaled versions of other vectors. Here, each scalar or coefficient influences how much the vector component contributes to forming the final target vector.

System of Equations

A system of equations is used when we have more than one equation, and we want to find values that satisfy all of them simultaneously. In this scenario, we have two equations derived from the components of vectors involved in the linear combination:

• \(-8c_1 + c_2 = -6\)
• \(4c_1 - c_2 = 4\)

These equations come from setting the linear combination equal to the desired vector's components.

Solving a system of equations can often involve methods like substitution, elimination, or using matrices. For this problem, we simplified the process by adding the equations to eliminate one variable, making it easier to solve for the others. This technique allowed us to solve the system step by step, ensuring we found the correct values for \(c_1\) and \(c_2\).

Constant Coefficients

Constant coefficients are the scalars that multiply the given vectors in a linear combination. In our exercise, coefficients \(c_1\) and \(c_2\) scale the vectors \(\langle -8, 4 \rangle\) and \(\langle 1, -1 \rangle\) respectively, to form the vector \(\langle -6, 4\rangle\).

These coefficients tell us how much of each vector is needed to achieve the desired linear combination. Finding them involves solving for \(c_1\) and \(c_2\) in a system of equations to satisfy the condition of the linear combination.

Once found, these coefficients provide the necessary scaling factors to recreate one vector using others. They are essential in applications like computer graphics, physics, and engineering, where objects and forces are often represented as vectors.