This simulator lets you dial in a vector's x, y and z components directly and watch the resultant arrow, its component staircase and its projection into the XY plane update together in a live 3D scene. Rotate the vector about the z axis or normalize it to unit length without ever conflating the two very different operations.
• A real-time 3D scene with home view, focus-selected-part, auto-rotate, expand and toggleable labels, tappable components with callouts (resultant vector arrow, orange/teal/violet component staircase, XY-plane projection with vertical connector, labeled coordinate axes, and endpoint marker). • Three controls: x component, y component and z component, each adjustable from −3 to 3 in 0.1 steps. • Play/pause, single-step (0.1 s) and larger-step (1 s) time controls, plus a playback-speed selector from 0.01× to 60× laboratory speed. • A rotate-about-z action, a normalize-vector action, and a pause-motion action, with a live sequence narrative describing what's happening. • Eight live metrics: magnitude |v|, XY projection length, unit-vector components ux/uy/uz, and direction angles α, β and γ in degrees. • A Curves & measurements tab with a direction-cosines chart, a projection-and-magnitude chart, the full model equations, and snapshot measurements. • An Experiments tab with four guided scenarios (a 3-4-5 scaled triangle, a pure vertical vector, normalizing without rotating, and a zero-vector diagnosis), a model-verification bench of independent automated checks, and a timestamped event log with a copyable trial report. • A Learn & assess tab with four guided lessons, a two-question knowledge-check quiz, and a written scope/reference statement linking to OpenStax's vectors in three dimensions chapter.
Each component is a signed displacement along one of three mutually perpendicular axes, and the magnitude |v| = √(vx²+vy²+vz²) collapses all three into a single nonnegative length — a negative component never implies a negative magnitude. The 3-4-5 experiment fixes vx=1.8, vy=2.4, vz=0 and shows the magnitude and the XY-projection length both land on exactly 3, since with vz=0 the whole vector already lies in that plane.
Set every component to zero and the magnitude readout drops to zero while the unit-vector components and all three direction angles become undefined — normalizing would require dividing by a zero magnitude, and there is no single direction that a zero-length vector can be said to point in. The endpoint marker deliberately stays visible even in this case so the origin itself remains a labeled point in the scene.
Normalizing a nonzero vector divides every component by its magnitude to produce a unit vector v̂ = v/|v|; this preserves direction exactly while forcing length to one, and is undefined only for the zero vector. Projecting onto the XY plane instead discards the z component entirely, producing projXY(v) = (vx,vy,0) — a length that can never exceed the full vector's length, illustrated by the pure-vertical experiment where setting only vz nonzero drives the XY projection to zero and the angle to +z to exactly 0°.
The direction cosines cos α = vx/|v|, cos β = vy/|v|, cos γ = vz/|v| are simply the components of the unit vector, and their squares always sum to one. Rotation in this lab is specifically about the z axis and preserves magnitude — it changes direction without changing length, the opposite trade-off from normalization.
No. Magnitude is defined as |v| = √(vx²+vy²+vz²), a square root of a sum of squares, so it is always a nonnegative length regardless of whether individual components are negative.
Normalizing divides each component by the magnitude, v̂ = v/|v|. For the zero vector, the magnitude is zero, so this division is undefined — there is also no single meaningful direction to assign to a vector with no length, which is why the unit-vector components and direction angles all become undefined at that point.
Projecting onto the XY plane discards the z component, producing (vx,vy,0) — a shorter (or equal-length) vector that keeps the original x and y values but loses all z information. Normalizing keeps the same direction as the original 3D vector but rescales its length to exactly one; it never changes which way the vector points.
They are the angles the vector makes with the positive x, y and z axes respectively. Their cosines equal the components of the unit vector (cos α = vx/|v|, and similarly for β and γ), and because the unit vector has length one, cos²α + cos²β + cos²γ always equals one.