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Scalar Projection of One Vector onto Another

compb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Units aren’t used in this calculation: every value is a plain number.

sig figs

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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See also

Dot Product from Magnitudes and Included Angle: Both use dot product and vector magnitude.Dot Product of Two 2D Vectors (Components): Both use dot product.Dot Product of Two 3D Vectors (Components): Both use dot product.Cross Product Magnitude: Both use vector magnitude.Magnitude of a 2D Vector: Both use vector magnitude.Magnitude of a 3D Vector: Both use vector magnitude.
5 more relatedMagnitude of a Scaled Vector: Both use vector magnitude.Resultant of Two Vectors at an Angle: Both use vector magnitude.Unit Vector Component (Normalization): Both use vector magnitude.x-Component from Magnitude and Angle: Both use vector magnitude.y-Component from Magnitude and Angle: Both use vector magnitude.
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Scalar Projection of One Vector onto Another explained

|b|compb a

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

compb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}
Where
  • compb⃗a⃗\text{comp}_{\vec{b}}\vec{a}= Scalar projection of a onto b
  • a⃗⋅b⃗\vec{a}\cdot\vec{b}= Dot product of a and b
  • ∣b⃗∣|\vec{b}|= Magnitude of b
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Vector operations

12 formulasMagnitude, direction, components and unit vectors, plus the dot and cross products and the projections and angles they give you.

Revised 31 October 2026
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formula.expert. (2026, October 31). Scalar Projection of One Vector onto Another Calculator. https://formula.expert/solvers/scalar-projection

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