Add and Subtract Coplanar Vectors

Introduction

Key Concepts

1. Understanding Vectors

A vector is a quantity that possesses both magnitude and direction, distinguishing it from scalar quantities, which have only magnitude. Examples of vectors include displacement, velocity, acceleration, and force. Vectors are typically represented graphically by arrows, where the length of the arrow denotes the magnitude and the arrowhead indicates the direction.

Mathematically, a vector can be expressed in terms of its components. In a two-dimensional plane, a vector **A** can be broken down into horizontal (x-axis) and vertical (y-axis) components: $$ \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} $$ where \( A_x \) and \( A_y \) are the magnitudes of the components along the x and y axes, respectively, and \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the x and y directions.

2. Coplanar Vectors

Coplanar vectors are vectors that lie within the same plane. Since they reside in a two-dimensional space, their addition and subtraction can be simplified using basic geometric principles. These vectors can be manipulated using graphical methods such as the head-to-tail method or algebraic methods involving their components.

3. Vector Addition

Vector addition involves combining two or more vectors to obtain a resultant vector. For coplanar vectors, this can be achieved using the head-to-tail method or by adding their respective components.

Head-to-Tail Method:

• Place the tail of the second vector at the head of the first vector.
• The resultant vector (**R**) is drawn from the tail of the first vector to the head of the second vector.

Component Method:

If vectors **A** and **B** have components \( A_x, A_y \) and \( B_x, B_y \) respectively, the resultant vector **R** is given by: $$ \mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \mathbf{i} + (A_y + B_y) \mathbf{j} $$

**Example:** Consider two vectors **A** and **B** where \( \mathbf{A} = 3\,\mathbf{i} + 4\,\mathbf{j} \) and \( \mathbf{B} = 1\,\mathbf{i} + 2\,\mathbf{j} \). The resultant vector **R** is: $$ \mathbf{R} = (3 + 1)\, \mathbf{i} + (4 + 2)\, \mathbf{j} = 4\,\mathbf{i} + 6\,\mathbf{j} $$

4. Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which can be interpreted as adding the negative of the second vector to the first. This operation is essential in determining the relative motion or force between two vector quantities.

Graphical Interpretation:

• To subtract vector **B** from vector **A** (**A** - **B**), reverse the direction of **B** to obtain **-B**.
• Add **-B** to **A** using the head-to-tail method to find the resultant vector.

Component Method:

If vectors **A** and **B** have components \( A_x, A_y \) and \( B_x, B_y \) respectively, the difference vector **D** is given by: $$ \mathbf{D} = \mathbf{A} - \mathbf{B} = (A_x - B_x) \mathbf{i} + (A_y - B_y) \mathbf{j} $$

**Example:** Using the previous vectors \( \mathbf{A} = 3\,\mathbf{i} + 4\,\mathbf{j} \) and \( \mathbf{B} = 1\,\mathbf{i} + 2\,\mathbf{j} \), the difference vector **D** is: $$ \mathbf{D} = (3 - 1)\, \mathbf{i} + (4 - 2)\, \mathbf{j} = 2\,\mathbf{i} + 2\,\mathbf{j} $$

5. Magnitude of the Resultant Vector

The magnitude of a resultant vector obtained from adding or subtracting vectors can be calculated using the Pythagorean theorem, provided the vectors are orthogonal (at right angles) or by using the law of cosines for vectors at an angle.

Orthogonal Vectors:

If vectors **A** and **B** are perpendicular, $$ |\mathbf{R}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2} $$

Vectors at an Angle:

If vectors **A** and **B** form an angle \( \theta \), $$ |\mathbf{R}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 + 2|\mathbf{A}||\mathbf{B}|\cos\theta} $$

**Example:** If **A** and **B** are at an angle of \( 60^\circ \), $$ |\mathbf{R}| = \sqrt{3^2 + 4^2 + 2 \times 3 \times 4 \times \cos60^\circ} = \sqrt{9 + 16 + 12} = \sqrt{37} \approx 6.08\, \text{units} $$

6. Direction of the Resultant Vector

The direction of the resultant vector can be determined using trigonometric ratios. The angle \( \phi \) the resultant makes with the x-axis is given by: $$ \phi = \tan^{-1}\left( \frac{R_y}{R_x} \right) $$ where \( R_x \) and \( R_y \) are the components of the resultant vector.

**Example:** Using the resultant vector **R** from the previous example \( \mathbf{R} = 4\,\mathbf{i} + 6\,\mathbf{j} \), $$ \phi = \tan^{-1}\left( \frac{6}{4} \right) = \tan^{-1}(1.5) \approx 56.31^\circ $$

7. Graphical Representation

Graphically representing vectors involves drawing arrows with lengths proportional to their magnitudes and in the given directions. The addition or subtraction of vectors can be visualized using the head-to-tail method, where vectors are placed sequentially, and the resultant is drawn from the starting point of the first vector to the endpoint of the last vector.

**Parallelogram Method:** For adding two vectors, **A** and **B**, a parallelogram is constructed with **A** and **B** as adjacent sides. The diagonal of the parallelogram represents the resultant vector **R**.

8. Algebraic vs. Graphical Methods

While graphical methods provide a visual understanding of vector addition and subtraction, algebraic methods offer precision and are more suitable for complex calculations. In physics problems where exact numerical answers are required, the component method is preferred.

However, graphical methods are invaluable for qualitative analysis and for providing intuitive insights into the behavior of vector quantities in physical scenarios.

Advanced Concepts

1. Vector Addition in Non-Orthogonal Systems

In many real-world applications, vectors do not align with the orthogonal axes, necessitating more advanced methods for their addition. When vectors are at arbitrary angles, the component method combined with trigonometric identities becomes essential.

Consider vectors **A** and **B** making angles \( \alpha \) and \( \beta \) with the x-axis, respectively. Their components are: $$ \mathbf{A} = A \cos\alpha \, \mathbf{i} + A \sin\alpha \, \mathbf{j} $$ $$ \mathbf{B} = B \cos\beta \, \mathbf{i} + B \sin\beta \, \mathbf{j} $$ The resultant vector **R** is: $$ \mathbf{R} = (A \cos\alpha + B \cos\beta) \, \mathbf{i} + (A \sin\alpha + B \sin\beta) \, \mathbf{j} $$

To find the magnitude and direction of **R**: $$ |\mathbf{R}| = \sqrt{(A \cos\alpha + B \cos\beta)^2 + (A \sin\alpha + B \sin\beta)^2} $$ $$ \phi = \tan^{-1}\left( \frac{A \sin\alpha + B \sin\beta}{A \cos\alpha + B \cos\beta} \right) $$

**Example:** Let \( \mathbf{A} = 5\,\text{N} \) at \( 30^\circ \) and \( \mathbf{B} = 7\,\text{N} \) at \( 120^\circ \).

Components: $$ \mathbf{A} = 5 \cos30^\circ \, \mathbf{i} + 5 \sin30^\circ \, \mathbf{j} = 4.33\,\mathbf{i} + 2.5\,\mathbf{j} $$ $$ \mathbf{B} = 7 \cos120^\circ \, \mathbf{i} + 7 \sin120^\circ \, \mathbf{j} = -3.5\,\mathbf{i} + 6.06\,\mathbf{j} $$

Resultant: $$ \mathbf{R} = (4.33 - 3.5)\, \mathbf{i} + (2.5 + 6.06)\, \mathbf{j} = 0.83\,\mathbf{i} + 8.56\,\mathbf{j} $$ $$ |\mathbf{R}| = \sqrt{0.83^2 + 8.56^2} \approx 8.59\,\text{N} $$ $$ \phi = \tan^{-1}\left( \frac{8.56}{0.83} \right) \approx 84.9^\circ $$

2. Vector Subtraction in Practical Applications

Vector subtraction is pivotal in various physics applications, such as determining relative velocities, forces, and accelerations. Understanding how to effectively subtract vectors can simplify complex problem-solving by breaking down forces or motions into manageable components.

**Example: Relative Velocity** Suppose a boat is moving with a velocity \( \mathbf{V}_b = 5\,\text{m/s} \) east, and there is a current with a velocity \( \mathbf{V}_c = 3\,\text{m/s} \) west. The resultant velocity \( \mathbf{V}_r \) of the boat relative to the shore is: $$ \mathbf{V}_r = \mathbf{V}_b - \mathbf{V}_c = 5\,\text{m/s} \, \mathbf{i} - 3\,\text{m/s} \, \mathbf{i} = 2\,\text{m/s} \, \mathbf{i} $$>

3. Mathematical Derivations

Deriving the formulas for vector addition and subtraction from first principles enhances the understanding of vector operations. Starting with the basic definition of vectors in component form, the addition and subtraction formulas naturally emerge from the combination of their respective components.

**Derivation of Resultant Magnitude:** Given two vectors **A** and **B**, the magnitude of the resultant vector **R** is: $$ |\mathbf{R}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2} $$> Expanding the squares: $$ |\mathbf{R}|^2 = (A_x)^2 + (B_x)^2 + 2A_x B_x + (A_y)^2 + (B_y)^2 + 2A_y B_y $$> Grouping terms: $$ |\mathbf{R}|^2 = (A_x^2 + A_y^2) + (B_x^2 + B_y^2) + 2(A_x B_x + A_y B_y) $$> Recognizing that \( A_x^2 + A_y^2 = |\mathbf{A}|^2 \) and \( B_x^2 + B_y^2 = |\mathbf{B}|^2 \): $$ |\mathbf{R}|^2 = |\mathbf{A}|^2 + |\mathbf{B}|^2 + 2|\mathbf{A}||\mathbf{B}|\cos\theta $$> where \( \theta \) is the angle between **A** and **B**. Taking the square root of both sides: $$ |\mathbf{R}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 + 2|\mathbf{A}||\mathbf{B}|\cos\theta} $$> This derivation underscores the relationship between the magnitudes of the vectors and the angle between them in determining the resultant.

4. Applications in Engineering and Technology

Vector addition and subtraction are extensively applied in engineering disciplines, including mechanics, electrical engineering, and aerospace. Understanding vector operations is crucial for analyzing force systems, designing structures, and optimizing performance in various technological applications.

**Structural Engineering:** Engineers use vector addition to determine the resultant forces acting on structures, ensuring stability and integrity.

**Aerospace Engineering:** In flight dynamics, vector operations help calculate resultant velocities, accelerations, and forces acting on aircraft and spacecraft.

**Electrical Engineering:** Vector addition is used in analyzing alternating current (AC) circuits, where voltages and currents are represented as phasors.

5. Challenges in Vector Operations

While vector addition and subtraction are foundational, several challenges may arise, especially when dealing with multiple vectors at various angles. Accurately decomposing vectors into components, ensuring correct sign conventions, and managing computational complexity are common hurdles.

**Solution Strategies:**

• Consistent Reference Axes: Establish a clear and consistent set of axes to simplify component calculations.
• Use of Technology: Employ calculators and software tools that handle vector computations efficiently.
• Practice with Diverse Problems: Strengthen understanding by solving a wide range of vector problems, enhancing proficiency and adaptability.

6. Interplay with Other Physics Concepts

Vector addition and subtraction are intrinsically linked to various other physics concepts, such as Newton's laws of motion, electromagnetism, and kinematics. Mastery of vector operations facilitates a deeper comprehension of these areas, enabling students to tackle complex physical phenomena with greater ease.

**Newton's First Law:** The equilibrium of forces acting on a body requires that the vector sum of all forces equals zero, necessitating precise vector subtraction to analyze force balances.

**Electric Fields:** The superposition principle in electromagnetism relies on the vector addition of electric fields from multiple charges to determine the resultant field at a point.

7. Vector Spaces and Advanced Mathematics

Delving into vector operations paves the way for exploring higher-dimensional vector spaces and linear algebra. Concepts such as vector bases, linear independence, and vector transformations extend the foundational knowledge of vector addition and subtraction into more abstract mathematical frameworks.

**Example:** In three-dimensional space, vectors include an additional z-component, complicating addition and subtraction. Understanding two-dimensional coplanar vectors is a stepping stone towards grasping these higher-dimensional operations.

Comparison Table

Summary and Key Takeaways