Finding Displacement Between Two Points

Introduction

Key Concepts

Understanding Displacement

Displacement refers to the change in position of an object from its initial point to its final point. Unlike distance, which is a scalar quantity representing the total path traveled, displacement is a vector quantity, meaning it has both magnitude and direction. This distinction is crucial in vector analysis, where direction plays a pivotal role in determining the resultant vector.

Position Vectors

A position vector is a vector that represents the position of a point in space relative to a chosen origin. In a two-dimensional Cartesian coordinate system, the position vector r of a point P(x, y) is given by:

$$ \mathbf{r} = x\,\mathbf{i} + y\,\mathbf{j} $$

Here, i and j are the unit vectors along the x-axis and y-axis, respectively. Position vectors are fundamental in defining displacement, as the displacement vector is derived from the difference between two position vectors.

Calculating Displacement

To find the displacement between two points, you subtract the initial position vector from the final position vector. Suppose an object moves from point A(x₁, y₁) to point B(x₂, y₂). The displacement vector d is calculated as:

$$ \mathbf{d} = \mathbf{r}_2 - \mathbf{r}_1 = (x_2 - x_1)\,\mathbf{i} + (y_2 - y_1)\,\mathbf{j} $$

The magnitude of the displacement vector, representing the straight-line distance between the two points, is determined using the Pythagorean theorem:

$$ |\mathbf{d}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Examples of Displacement Calculation

**Example 1:** An object moves from point A(2, 3) to point B(5, 7). Calculate the displacement vector and its magnitude.

Solution:

Given:

• r₁ = 2i + 3j
• r₂ = 5i + 7j

Displacement vector:

$$ \mathbf{d} = \mathbf{r}_2 - \mathbf{r}_1 = (5 - 2)\,\mathbf{i} + (7 - 3)\,\mathbf{j} = 3\,\mathbf{i} + 4\,\mathbf{j} $$

Magnitude of displacement:

$$ |\mathbf{d}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

Therefore, the displacement vector is 3i + 4j with a magnitude of 5 units.

Displacement in Three Dimensions

While the previous examples focus on two-dimensional space, displacement can also be calculated in three dimensions. For a point P(x, y, z), the position vector is:

$$ \mathbf{r} = x\,\mathbf{i} + y\,\mathbf{j} + z\,\mathbf{k} $$

Given two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂), the displacement vector is:

$$ \mathbf{d} = \mathbf{r}_2 - \mathbf{r}_1 = (x_2 - x_1)\,\mathbf{i} + (y_2 - y_1)\,\mathbf{j} + (z_2 - z_1)\,\mathbf{k} $$

And its magnitude is:

$$ |\mathbf{d}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} $$>

Applications of Displacement Vectors

Displacement vectors are widely used in various fields, including physics, engineering, and computer graphics. In physics, they help describe the motion of objects, enabling the calculation of velocity and acceleration. Engineers use displacement vectors in structural analysis and design, ensuring that components move as intended under different forces. In computer graphics, displacement vectors are essential for rendering movement and animations.

Graphical Representation of Displacement

Displacement vectors can be visually represented on a coordinate system. By plotting the initial and final points, the displacement vector is drawn as an arrow from the initial point to the final point. The length of the arrow corresponds to the magnitude of displacement, while the direction of the arrow indicates the direction of movement.

**Example:** Plotting the displacement from point A(1, 2) to point B(4, 6) involves drawing a vector from (1, 2) to (4, 6), resulting in the displacement vector 3i + 4j.

Component Form of Vectors

Vectors can be expressed in their component forms, which simplifies calculations involving displacement. A displacement vector d in two dimensions can be broken down into its horizontal (d_x) and vertical (d_y) components:

$$ \mathbf{d} = d_x\,\mathbf{i} + d_y\,\mathbf{j} $$>

Understanding the component form is essential when adding or subtracting vectors, as it allows for the independent manipulation of each component.

Vector Addition and Displacement

Displacement vectors can be added together using vector addition principles. If an object undergoes multiple displacements, the resultant displacement is the vector sum of all individual displacements.

Example: An object moves from A(0, 0) to B(3, 4), then to C(6, 8). The total displacement from A to C is:

$$ \mathbf{d}_{AC} = \mathbf{d}_{AB} + \mathbf{d}_{BC} = (3\,\mathbf{i} + 4\,\mathbf{j}) + (3\,\mathbf{i} + 4\,\mathbf{j}) = 6\,\mathbf{i} + 8\,\mathbf{j} $$>

The magnitude of d_AC is:

$$ |\mathbf{d}_{AC}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $$>

Unit Vectors and Direction Cosines

Unit vectors are vectors with a magnitude of one, indicating direction. They are useful in expressing displacement vectors in terms of directional components. The direction cosines of a vector define the angles between the vector and the coordinate axes and are essential in three-dimensional displacement calculations.

For a displacement vector d with components d_x, d_y, d_z, the direction cosines are:

$$ \cos(\alpha) = \frac{d_x}{|\mathbf{d}|}, \quad \cos(\beta) = \frac{d_y}{|\mathbf{d}|}, \quad \cos(\gamma) = \frac{d_z}{|\mathbf{d}|} $$>

Where α, β, γ are the angles between the vector and the x, y, z-axes, respectively.

Displacement and Motion Equations

Displacement is integral to the kinematic equations that describe motion. For uniformly accelerated motion, displacement can be calculated using the following equation:

$$ \mathbf{d} = \mathbf{v}_i t + \frac{1}{2} \mathbf{a} t^2 $$>

Where:

• v_i is the initial velocity.
• a is the acceleration.
• t is the time elapsed.

Relative Displacement

Relative displacement considers the position of an object relative to another moving object. It is particularly useful in scenarios where multiple reference frames are involved. By analyzing relative displacement, one can determine the motion of one object as observed from another.

Example: If car A moves east at 60 km/h and car B moves west at 40 km/h, the relative displacement of car B with respect to car A can be calculated by vector subtraction of their velocities.

Displacement in Polar Coordinates

While Cartesian coordinates are common, displacement can also be expressed in polar coordinates, especially in rotational motion contexts. In polar coordinates, a point is defined by its radial distance r and angle θ. The displacement vector in polar coordinates is:

$$ \mathbf{d} = r\,\mathbf{e}_r + r\theta\,\mathbf{e}_θ $$>

Where e_r and e_θ are the radial and angular unit vectors, respectively.

Displacement in Physics

In physics, displacement is fundamental in describing motion. It is distinct from other measures like speed and velocity because it accounts for the direction of movement. Understanding displacement helps in analyzing forces, energy, and momentum within a system.

Displacement and Work Done

Work done by a force is directly related to displacement. The relationship is given by the equation:

$$ W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}| |\mathbf{d}| \cos(\theta) $$>

Where:

• W is the work done.
• F is the force applied.
• d is the displacement.
• θ is the angle between the force and displacement vectors.

This equation highlights the importance of both the magnitude and direction of displacement in determining the work done by a force.

Graphical Methods for Finding Displacement

Graphical methods, such as vector addition diagrams, are effective tools for visualizing displacement. By representing vectors graphically, students can better understand the direction and magnitude relationships between different displacement vectors.

Steps for Graphical Construction:

1. Plot the initial point on a coordinate system.
2. From this point, draw the first displacement vector to its endpoint.
3. From the endpoint of the first vector, draw the second displacement vector.
4. Continue this process for all displacement vectors.
5. The resultant displacement vector is drawn from the initial point to the final endpoint.

Displacement in Real-Life Scenarios

Displacement vectors are applicable in numerous real-life situations. For instance, in navigation, displacement helps in determining the shortest path between two locations. In sports, analyzing an athlete's displacement can improve performance strategies. Additionally, in robotics, displacement vectors guide the movement and positioning of robotic limbs.

Challenges in Understanding Displacement

Students often confuse displacement with distance due to their similar nature. Emphasizing the vector aspect of displacement—its direction and magnitude—is essential for clarity. Additionally, working with vectors in three-dimensional space can be challenging, requiring a strong grasp of vector components and operations.

Advanced Topics: Displacement in Non-Cartesian Coordinates

While Cartesian coordinates are widely used, displacement can also be analyzed in other coordinate systems, such as cylindrical or spherical coordinates. These systems are particularly useful in fields like electromagnetism and quantum mechanics, where spherical symmetry is prevalent.

Comparison Table

Aspect	Displacement	Distance
Definition	Vector quantity representing the change in position from initial to final point.	Scalar quantity representing the total path traveled.
Direction	Has both magnitude and direction.	Has only magnitude; no direction.
Formula	d = r2 - r1	Sum of all incremental distances traveled.
Example	Moving from (2,3) to (5,7) has a displacement vector of 3i + 4j.	Traveling along a path covering 10 units in total.
Applications	Used in physics for motion analysis, engineering for design, and computer graphics for movement.	Used to measure the length of a journey or route.
Pros	Provides precise information about position change, essential for vector analysis.	Easy to calculate for simple paths.
Cons	Requires understanding of vectors and directional components.	Does not provide information about direction, limiting its use in vector-based applications.

Summary and Key Takeaways