How does the resultant of two vectors change as the angle between the two vectors increase?
To understand how the resultant of two vectors changes as the angle between them increases, we first need to establish a few concepts.
When two vectors are added together, their resultant is determined by the vector addition process. This can be done graphically or mathematically.
Graphically, you can draw the two vectors tail-to-tail and connect the free ends to form a triangle. The resultant vector is the vector that starts from the tail of the first vector and ends at the tip of the second vector.
Mathematically, you can use vector components. If you have two vectors A and B, their components can be written as (Ax, Ay) and (Bx, By) respectively. The resultant vector R can be found by adding the corresponding components: R = (Ax + Bx, Ay + By).
Now, let's consider how the angle between two vectors affects their resultant:
1. When the angle between two vectors is 0 degrees (or they are collinear), the resultant vector is the vector sum of the magnitudes of the two vectors. This is the maximum value the resultant can attain.
2. When the angle between the two vectors is 180 degrees (or they are collinear but in opposite directions), the resultant vector is the vector difference of the magnitudes of the two vectors. The resultant will be the difference between the two magnitudes and can range from the maximum value to the minimum value.
3. As the angle between the two vectors increases from 0 degrees to 180 degrees (or anywhere in between), the magnitude of the resultant vector decreases. This is because the components of the vectors start to oppose each other in some way, contributing less to the overall magnitude of the resultant.
4. Additionally, as the angle between the two vectors increases, the direction of the resultant vector changes. It rotates gradually from aligning with one of the vectors towards aligning with the other vector.
So, in summary, as the angle between two vectors increases, the magnitude of the resultant vector decreases, and its direction also changes. The resultant reaches its minimum value when the two vectors are in opposite directions and its maximum value when they are collinear.