The resultant of two displacement vectors having the same direction is the sum of the two displacements having the same direction as the original vectors.
When two displacement vectors point in the same direction, combining them is straightforward. Unlike vectors pointing in different directions, which require more complex methods like the parallelogram rule or vector components, vectors aligned in the same direction simply add up.
Understanding Displacement Vectors
A displacement vector represents the change in position of an object. It has both magnitude (how far the object moved) and direction (in which direction it moved). For example, walking 5 meters East is a displacement vector. Walking another 3 meters East from that point is a second displacement vector.
Calculating the Resultant
As stated in the reference, when two displacement vectors share the exact same direction:
- The magnitude of the resultant vector is the sum of the magnitudes of the individual vectors.
- The direction of the resultant vector is the same as the direction of the original vectors.
This is because the displacements are cumulative and extend along the same line.
Example Calculation
Let's consider two displacement vectors:
- Vector A: 10 meters North
- Vector B: 5 meters North
Both vectors point in the same direction (North).
To find the resultant vector R:
- Add the magnitudes: Magnitude of R = Magnitude of A + Magnitude of B = 10 m + 5 m = 15 meters.
- Keep the direction: The direction of R is the same as A and B, which is North.
So, the resultant displacement is 15 meters North.
Visualizing the Resultant
Imagine drawing these vectors head-to-tail. If Vector A is drawn as an arrow 10 units long pointing North, and Vector B is drawn starting from the tip of Vector A as an arrow 5 units long also pointing North, the combined path forms a single arrow 15 units long pointing North.
Key Properties
Here's a summary of the properties when combining two displacement vectors with the same direction:
| Property | Result |
|---|---|
| Magnitude | Sum of individual magnitudes |
| Direction | Same as the direction of the individual vectors |
| Operation | Simple addition (scalar addition of magnitudes) |
This principle holds true for any number of vectors acting along the same line and in the same direction.
Practical Insights
This concept is fundamental in physics and everyday life:
- If you walk 50 meters East and then another 30 meters East, your total displacement is 80 meters East.
- If a car travels 100 km North and then continues for another 50 km North, its total displacement is 150 km North.
- In navigation, if a ship sails 20 nautical miles West and then 15 nautical miles further West, its total displacement is 35 nautical miles West.
In each case, the total effect of the movements in the same direction is simply the sum of the individual movements, maintaining the original direction. The resultant vector encapsulates the net change in position from the start to the end point.
