Introduction:

Understanding the concept of resultant displacement is crucial in various fields, including physics, engineering, and navigation. This guide will explore how to calculate the resultant displacement of a body that moves 6 meters north, 8 meters east, and 10 meters vertically upwards. We will break down the process using vector calculation and the Pythagorean theorem, providing a detailed explanation and practical examples.

Understanding Displacement and Vectors

Displacement is a vector quantity that describes the shortest distance from the initial position to the final position of a body. It takes into account the direction of the movement. In a three-dimensional space, displacement is represented as a vector with components along the x, y, and z axes.

Vector Representation

Let's consider the movements of a body in a three-dimensional space:

North movement: A (0, 6, 0) East movement: B (8, 0, 0) Upward movement: C (0, 0, 10)

These movements can be represented as vectors:

A [begin{pmatrix}0 6 0end{pmatrix}], meters

B [begin{pmatrix}8 0 0end{pmatrix}], meters

C [begin{pmatrix}0 0 10end{pmatrix}], meters

Summing the Vectors

The resultant vector D can be calculated by summing these vectors:

[vec{D} vec{A} vec{B} vec{C} begin{pmatrix}8 6 10end{pmatrix}] meters

Calculating the Magnitude of the Resultant Vector

The magnitude of the resultant displacement vector (vec{D}) can be calculated using the formula for the magnitude of a vector in three-dimensional space:

[D sqrt{x^2 y^2 z^2}]

Substituting the values:

[D sqrt{8^2 6^2 10^2} sqrt{64 36 100} sqrt{200} 10 sqrt{2} approx 14.14 , text{m}

Therefore, the resultant displacement from the initial position is approximately 14.14 meters.

Alternative Methods Using the Pythagorean Theorem

Another way to calculate the resultant displacement is by using the Pythagorean theorem. This method involves breaking the problem into two steps:

Calculate the displacement at ground level using the north and east movements. Use the Pythagorean theorem again to determine the final resultant displacement in 3D space.

Displacement at Ground Level

The north and east movements form a right-angled triangle. Using the Pythagorean theorem:

[text{Displacement at ground level} sqrt{6^2 8^2} sqrt{36 64} sqrt{100} 10 , text{m}

Final Resultant Displacement

The vertical displacement of 10 meters, combined with the 10 meters displacement at ground level, forms another right-angled triangle. Applying the Pythagorean theorem again:

[text{Final resultant displacement} sqrt{10^2 10^2} sqrt{100 100} sqrt{200} 10 sqrt{2} approx 14.14 , text{m}

Advanced Considerations: Non-Euclidean Geometry

Euclidean Geometry: In a "flat" space, the resultant displacement can be calculated using simple vector addition and the Pythagorean theorem. This scenario is simplified and commonly used in most practical applications.

Non-Euclidean Geometry: However, in curved spaces, such as on a sphere or an ellipsoid, the calculations become more complex. In a spherical geometry, if a body moves 6 meters north and 8 meters east on a sphere with a circumference of 50 meters, the movements are more complex due to the curvature of the space. The displacement in such a scenario can vary significantly.

Illustrative Example: Sphere with a Circumference of 50m

Consider a sphere with a circumference of 50 meters. If you start at the south pole and move 6 meters north, you will reach the north pole. Moving 8 meters east at the north pole means you are spinning in place. Finally, moving 10 meters vertically upwards:

Moving 25 meters north brings you to the north pole. Spinning east only displaces you vertically due to the curvature. The vertical displacement can be calculated as: [text{Displacement} 50 / pi times 30 approx 45.915 , text{meters}

The final resultant displacement can be a maximum of 55 meters due to the curvature of the sphere.

Conclusion: Calculating resultant displacement is a fundamental concept in three-dimensional space. Whether using vector addition or the Pythagorean theorem, the method can vary based on whether the space is flat or curved. Understanding these principles is essential for accurate calculations in various scientific and engineering applications.