How this instrument works
A vector is fully described by two separate pieces of information: its magnitude (how long it is) and its direction (which way it points). Given its x and y components, the magnitude follows from the Pythagorean theorem, |v| = √(x²+y²), and the direction follows from the two-argument arctangent, angle = atan2(y,x) — the same relationships this site's separate Vector Magnitude and Vector Direction pages compute individually, brought together here for whenever both are wanted at once.
Splitting a vector into these two pieces is useful because many calculations only care about one half — normalizing a vector to a unit length only needs the magnitude, while comparing two vectors' headings only needs the direction — but a vector's FULL description genuinely requires both together, exactly what this combined page returns in a single pass.
The zero vector, (0, 0), is the one case where this description breaks down: its magnitude is a well-defined zero, but its direction is meaningless, since a vector with no length doesn't point anywhere in particular. Every other vector has both a well-defined length and a well-defined heading.
- Enter the vector's horizontal component into the x field.
- Enter the vector's vertical component into the y field.
- Read Magnitude and Direction: the sheet computes both simultaneously from the same two components.
Worked example — the vector (3, 4)
The vector (3, 4) has a magnitude of 5 and a direction of approximately 53.13° — a full description reached in one pass, rather than checking the separate magnitude-only and direction-only pages elsewhere on this site one at a time.
The zero vector, (0, 0), has a magnitude of exactly 0 and no meaningful direction, since a vector with no length doesn't point anywhere. The vector (5, 12) instead has a magnitude of 13 (the classic 5-12-13 triple) and a direction of about 67.38°.
Questions
What two numbers describe a vector fully?
Its magnitude (length) and its direction (which way it points). Both are needed together for a complete description — knowing only one leaves the vector's full picture incomplete.
How is this different from the separate magnitude and direction pages?
Those pages each return one half of a vector's description on their own. This page computes both simultaneously from the same two components, for whenever the full picture is wanted in a single pass.
What is the direction of the zero vector?
Undefined — a vector with no length at all doesn't point anywhere in particular, since direction only makes sense for something that actually extends some distance from the origin.
Why is atan2 used instead of a plain arctangent for the direction?
A plain arctangent only sees the ratio y⁄x, which is identical for a vector and its exact opposite. atan2 reads both components' individual signs and places the direction in the correct quadrant automatically.
Can a vector's magnitude be negative?
No — magnitude is always zero or positive, since it comes from a square root, which by convention returns the non-negative result. A vector can point in a negative direction along an axis, but its overall length is never itself negative.