Vectors: Dot & Cross
3D vector playground.
Two forces act on an object at the same time. One force is 5 Newtons pushing East. The other is 5 Newtons pushing North.
What is the magnitude of the TOTAL force on the object?
Forces are vectors — they have both magnitude AND direction. When two perpendicular forces combine, use the Pythagorean theorem: √(5² + 5²) = √50 ≈ 7.07 N, pointing northeast at 45°.
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Drag the vectors to different angles. Watch how the resultant changes.
|A + B|² = |A|² + |B|² + 2A·B
When A and B are perpendicular, A·B = 0, and this becomes the Pythagorean theorem!
A pilot must add the wind vector to their airspeed vector. A 30 km/h crosswind + 200 km/h airspeed ≠ 230 km/h groundspeed — the pilot drifts sideways.
Every game engine uses vectors for movement, forces, and camera angles. The dot product detects if enemies face you; the cross product computes surface normals.
Your phone measures acceleration as a 3D vector. Tilt = gravity vector changing direction relative to the phone. The magnitude stays 9.8 m/s².
“Vectors are the alphabet of physics. Once you think in vectors, you stop asking 'how much?' and start asking 'how much in which direction?'”
Vector Challenge
Given a target vector, combine two adjustable vectors to match it. A vector puzzle!
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Can you hit the target using two vectors at right angles? What about two parallel vectors?
Any vector in 2D can be decomposed into components along ANY two non-parallel directions. The x-y axes are just the most convenient choice.