2D vector sum via the triangle and parallelogram rules
📘 Conceptual notes
A boater wants to cross a river. The boat has its own velocity, but the current pushes it sideways. The upshot: the boat doesn't go where it's pointed — it drifts.
Play with the sliders and switch on the layers to see how the vector sum determines the boat's actual path.
▶ What each vector represents vec_vb · vec_vc
Two independent velocities acting at the same time:
- — the boat's velocity relative to the water. It's what the motor (or the oar) produces. With no current, the boat would head exactly that way.
- — the current's velocity. Acts horizontally, pushing everything in the water downstream.
The angle is the direction the pilot picks for the bow. Move the slider and see how it changes the outcome.
▶ The resultant: where the boat actually goes vec_vR
From the bank, you see the boat travel along a third direction — neither the motor's nor the current's, but the sum of the two:
Switch on the resultant layer and watch: the stronger the current relative to the boat, the more it's pulled off course.
▶ Decomposing into components components
To compute , we split into two pieces — one horizontal (along the river) and one vertical (crossing the river):
The current only acts horizontally, so the resultant becomes:
The magnitude follows from Pythagoras:
▶ Drift angle: how much the boat deviates angle_alpha
The angle measures the deviation between the current's direction and the boat's actual direction. In practical terms: the larger , the more the boat manages to "fight" the current.
Try it: if and , the boat crosses in a straight line. If , even heading straight against the current it can't make it — it'll always be swept downstream.
▶ ✈️ Analogy: a plane with a crosswind
Same problem, different setting. A pilot points the nose at the destination, but a crosswind pushes the plane sideways. Seen from the ground, the trajectory is oblique.
The fix is identical to the boater's: aim partly into the wind to cancel the drift. The physics underneath is the same vector sum.
▶ 🧭 Trajectory vs. displacement
The red trail is the trajectory — every point along the path travelled. The dashed green line is the displacement — it connects the origin to the current point in a straight line.
If you don't change anything during the trip, the two coincide. But if you change or the current mid-journey, the trajectory bends while the displacement stays straight.