Magnitude of a 2D Vector
Worked example: Components (3, 4) → magnitude 5 — press Try an example to run it live, then adjust anything.
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Grade 12Grade 12 Math
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Magnitude of a 2D Vector explained
A vector is an arrow: a direction plus a length. Drop it onto axes and it splits into two ordinary numbers, the components vx and vy, which form the legs of a right triangle whose hypotenuse is the arrow itself — so the magnitude is √(vx² + vy²). A hiker who walks 3 km east and 4 km north ends up 5 km from camp, not 7 km, because displacement is a vector and its parts add head-to-tail rather than arithmetically. That gap between 5 and 7 is the single most common student error in the whole subject.
Run it backwards and a magnitude plus one component recovers the other: a vector of length 13 with vy = 5 must have vx = √(169 − 25) = 12. The solver returns the positive root, since a component's sign is a matter of which way the axis points and the arithmetic cannot know it. Note also that no component may exceed the magnitude — a leg longer than the hypotenuse describes a triangle that does not exist.
Magnitude of a 2D Vector formula
- = Vector magnitude
- = x-component
- = y-component
Missing one of these? Work it out first, then come back
- Vector magnitude — Magnitude of a 3D Vector, Unit Vector Component (Normalization)
- x-component — Magnitude of a 3D Vector, Dot Product of Two 2D Vectors (Components)
- y-component — Magnitude of a 3D Vector, Dot Product of Two 2D Vectors (Components)