Magnitude and Direction

Calculus 2 · Axiom Academy

A plane flies one way, the wind pushes another — and the path it actually traces over the ground is the two velocities added as vectors. You're flying due East at 200 mph through the air , but the air itself is moving. A velocity isn't just a number — it carries a magnitude and a direction , and that's exactly what lets you combine the plane and the wind into your true ground track. Your engine points the plane due East at 200 mph. Now turn on the wind — set its speed and the way it blows — and watch your real path over the ground swing away from where you're pointed. Ground velocity = air velocity + wind velocity, added tip-to-tail. Split a velocity into East & North To add or compute with vectors, you break each one into its East and North parts. Drag the heading of this 50 mph velocity and watch its two shadows: the East leg is 50 cosθ, the North leg is 50 sinθ. Square them, add, square-root — and you get the 50 right back. Here's the pilot's real question: that 40 mph wind never changes its speed, only its direction. Spin it all the way around and watch your ground speed swing from a 240 mph push to a 160 mph fight — with a pure sideways crosswind landing in between. Direction is the whole story.

This is the written version of the interactive lesson above. See the full Calculus 2 course.