Scalar Projection of One Vector onto Another
Worked example: dot = 24 onto a vector of length 6 → projection 4 — press Try an example to run it live, then adjust anything.
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Grade 12Grade 12 Math
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Scalar Projection of One Vector onto Another explained
Shine a light perpendicular to b and the shadow that a casts along it has length (a·b)/|b| — equivalently |a| cos θ. This is how a diagonal quantity gets charged to a chosen direction: the component of a ship's velocity along the intended course, the part of a rope's tension that actually lifts, the useful fraction of an angled push. A dot product of 24 against a vector b of length 6 means a reaches 4 units along b's direction, regardless of how long a itself is or how far it strays sideways.
The trap sits in the denominator. It is |b|, the vector being projected onto, never |a|. Dividing by the wrong magnitude quietly returns the cosine of the angle scaled by the wrong length, and the mistake survives dimensional analysis, so it will not announce itself. A negative result is perfectly legitimate and means a leans backwards along b — its shadow falls on the far side of the origin.
Scalar Projection of One Vector onto Another formula
- = Scalar projection of a onto b
- = Dot product of a and b
- = Magnitude of b
Missing one of these? Work it out first, then come back
- Dot product of a and b — Dot Product of Two 2D Vectors (Components), Dot Product of Two 3D Vectors (Components)
- Magnitude of b — Dot Product from Magnitudes and Included Angle, Cross Product Magnitude