Problem 27 - Fundamentals of Physics

Question

If \(\vec{d}_{1}+\vec{d}_{2}=5 \vec{d}_{3}, \vec{d}_{1}-\vec{d}_{2}=3 \vec{d}_{3}\), and \(\vec{d}_{3}=2 \hat{i}+4 \hat{j}\), then what are, in unit-vector notation, (a) \(\vec{d}_{1}\) and (b) \(\vec{d}_{2}\) ?

Step-by-Step Solution

Answer

\( \vec{d}_1 = 8 \hat{i} + 16 \hat{j} \), \( \vec{d}_2 = 2 \hat{i} + 4 \hat{j} \).

Step 1: Set Up Equations

We are given two vector equations: 1. \( \vec{d}_1 + \vec{d}_2 = 5 \vec{d}_3 \) 2. \( \vec{d}_1 - \vec{d}_2 = 3 \vec{d}_3 \) We also know that \( \vec{d}_3 = 2 \hat{i} + 4 \hat{j} \). We start by substituting \( \vec{d}_3 \) into the equations.

Step 2: Substitute Values of \\vec{d}_3

Substitute \( \vec{d}_3 = 2 \hat{i} + 4 \hat{j} \) into the two equations:\[ \vec{d}_1 + \vec{d}_2 = 5(2 \hat{i} + 4 \hat{j}) = 10 \hat{i} + 20 \hat{j} \]\[ \vec{d}_1 - \vec{d}_2 = 3(2 \hat{i} + 4 \hat{j}) = 6 \hat{i} + 12 \hat{j} \]

Step 3: Add Equations to Solve for \\vec{d}_1

Add the equations to eliminate \( \vec{d}_2 \):\[ (\vec{d}_1 + \vec{d}_2) + (\vec{d}_1 - \vec{d}_2) = (10 \hat{i} + 20 \hat{j}) + (6 \hat{i} + 12 \hat{j}) \]\[ 2\vec{d}_1 = 16 \hat{i} + 32 \hat{j} \] Divide by 2:\[ \vec{d}_1 = 8 \hat{i} + 16 \hat{j} \]

Step 4: Plug Back to Find \\vec{d}_2

Substitute \( \vec{d}_1 = 8 \hat{i} + 16 \hat{j} \) back into the first equation to find \( \vec{d}_2 \):\[ 8 \hat{i} + 16 \hat{j} + \vec{d}_2 = 10 \hat{i} + 20 \hat{j} \]Solving for \( \vec{d}_2 \) gives:\[ \vec{d}_2 = (10 \hat{i} + 20 \hat{j}) - (8 \hat{i} + 16 \hat{j}) \]\[ \vec{d}_2 = 2 \hat{i} + 4 \hat{j} \]

Step 5: Conclude the Solution

Now that we've solved for both vectors, summarize the solution:- \( \vec{d}_1 = 8 \hat{i} + 16 \hat{j} \)- \( \vec{d}_2 = 2 \hat{i} + 4 \hat{j} \)

Key Concepts

Unit-Vector Notation

Understanding unit-vector notation is crucial when working with vectors, especially in physics and engineering. In unit-vector notation, vectors are broken down into their component directions, using standard unit vectors.

• Unit Vectors: These are vectors with a magnitude of one, commonly denoted as \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \). In the Cartesian coordinate system, \( \hat{i} \) represents the x-axis direction, \( \hat{j} \) represents the y-axis direction, and \( \hat{k} \) represents the z-axis direction.
• Component Representation: Any vector in the plane (2D) can be expressed as a combination of these unit vectors. For example, if you have \( \vec{v} = a \hat{i} + b \hat{j} \), it means the vector has a magnitude of \( a \) in the x-direction and \( b \) in the y-direction.

Unit-vector notation simplifies the process of vector addition, as you only need to focus on adding the components along each axis. In our exercise, representing \( \vec{d}_3 \) as \( 2\hat{i} + 4\hat{j} \) allows straightforward calculation in finding both \( \vec{d}_1 \) and \( \vec{d}_2 \). This notation keeps calculations neat and helps visualize direction and magnitude easily.

Vector Equations

Vector equations provide a powerful method for solving problems involving multiple vectors. They allow you to work with vectors algebraically in the same way you would with numbers.

• Setting Up Equations: To solve problems, it's important to first express all your known and unknown vectors using equations. In this case, the two equations are \( \vec{d}_1 + \vec{d}_2 = 5 \vec{d}_3 \) and \( \vec{d}_1 - \vec{d}_2 = 3 \vec{d}_3 \).
• Substitution: By substituting known vectors into these equations, you reduce the problem to algebraic manipulation. For our exercise, substituting \( \vec{d}_3 \) as \( 2\hat{i} + 4\hat{j} \) allows you to solve these equations step-by-step.

Vector equations streamline the process of isolating variables and solving for unknown vectors. By comparing coefficients from unit-vector components, solving these becomes practically systematic. This method was used in the exercise to individually determine \( \vec{d}_1 \) and \( \vec{d}_2 \).

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and operations like vector addition. It's essential for understanding how vectors interact in various fields such as physics and engineering.

• Vector Addition: This is a fundamental operation where corresponding components of vectors are added together. It forms the basis of the vector equations seen in our exercise.
• Linear Combinations: A core concept, where vectors are combined linearly using scalar coefficients, like in \( 5 \vec{d}_3 \) and \( 3 \vec{d}_3 \) seen in the equations. These coefficients scale the vectors, contributing to their sum or difference.
• Solving Systems: Linear algebra often involves solving systems of equations, typically involving sets of vector equations. Using algebraic methods, you can find solutions to these equations efficiently.

In our exercise, the principles of linear algebra help structure the steps to isolate \( \vec{d}_1 \) and \( \vec{d}_2 \). By utilizing these algebraic methods, the solution is derived systematically, illustrating the power and utility of linear algebra in solving vector problems.