The final distance in a sequence refers to the total length covered after following a series of movements or steps. This could involve moving in straight lines or along a path with multiple segments. Understanding how to calculate this distance is crucial in various fields, from navigation to physics.
Calculating Final Distance in a 2D Plane
When dealing with points in a 2D plane, the distance between two points
$(x_1, y_1)$ and $(x_2, y_2)$ can be calculated using the distance formula:
$d = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Example
Imagine you start at point A (2, 3) and move to point B (5, 7). The distance between these points is:
$d = sqrt{(5 – 2)^2 + (7 – 3)^2} = sqrt{3^2 + 4^2} = sqrt{9 + 16} = sqrt{25} = 5$
Summing Distances in a Sequence
If you have multiple segments, you simply add the distances together. For instance, if you move from A to B, then from B to C, you sum the distances from A to B and B to C.
Example
Suppose you move from A (2, 3) to B (5, 7), and then from B (5, 7) to C (8, 10). First, calculate the distance from A to B:
$d_{AB} = 5$
Next, calculate the distance from B to C:
$d_{BC} = sqrt{(8 – 5)^2 + (10 – 7)^2} = sqrt{3^2 + 3^2} = sqrt{9 + 9} = sqrt{18} = 3sqrt{2}$
Finally, sum these distances to get the total distance:
$d_{total} = d_{AB} + d_{BC} = 5 + 3sqrt{2}$
Calculating Final Distance in 3D Space
In three-dimensional space, the distance formula extends to include the z-coordinates. For points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, the formula is:
$d = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}$
Example
If you move from point A (1, 2, 3) to point B (4, 6, 8), the distance is:
$d = sqrt{(4 – 1)^2 + (6 – 2)^2 + (8 – 3)^2} = sqrt{3^2 + 4^2 + 5^2} = sqrt{9 + 16 + 25} = sqrt{50} = 5sqrt{2}$
Conclusion
Understanding how to find the final distance in a sequence of movements or steps is essential in many real-world applications. Whether you’re dealing with points on a map or navigating through three-dimensional space, the distance formula provides a reliable method for calculating the total distance covered.