Vector Projections

Calculus 2 · Axiom Academy

The shadow one vector casts along another — and how that single idea splits any vector into a parallel part and a perpendicular part. Place and tail-to-tail. Drop light straight down onto the line of . Where the tip of lands is the foot of the shadow, and the arrow from the origin to that foot is the vector projection . The dashed drop meets at a right angle. 2. The Scalar Component — a Signed Length Before the vector, ask for just the length of that shadow: the scalar component . It's one number. Watch the readout build it as the foot slides out, then watch what happens when leans backward past — the number turns negative . 3. Turning the Length Back Into a Vector To get a vector , point that length along : multiply the scalar component by the unit vector . Stacking three copies of marches exactly to the foot at . Folding together gives the compact form . scalar length × the direction of the compact, ready-to-compute form 4. Splitting u Into Parallel + Perpendicular The leftover, , is the part of that has no shadow — it stands perpendicular to . Every vector splits this way: a piece along (the projection) plus a piece perpendicular to . Watch break into its green and purple legs, then reassemble. — runs along . In physics this is the part of a force that does work along the motion ( ). — at a right angle to . It's the "wasted" sideways component, like the part of gravity pressing into a ramp.

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