Dot Product & Angle Between Vectors

Inputs

u · v = ac + bd | cos θ = (u·v) / (|u| |v|)

Vector u = ⟨a, b⟩

u =

a (x-component)

b (y-component)

Vector v = ⟨c, d⟩

v =

c (x-component)

d (y-component)

Load an example:

Results

Enter components for u = ⟨a, b⟩ and v = ⟨c, d⟩, then press Compute to find the dot product, angle, and projection.

Dot Product u · v

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|u| — Magnitude of u

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|v| — Magnitude of v

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cos θ

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Angle θ (degrees)

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Angle θ (radians)

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Relationship

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Scalar Projection comp_v(u) = u·v / |v|

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Vector Projection proj_v(u)

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Dot Product — Formula & Geometric Meaning

The dot product (scalar product) of two vectors combines their components into a single number:

Orthogonality test: u · v = 0 means the vectors are perpendicular (cos 90° = 0).

Parallel test: |cos θ| = 1 means the vectors point in the same or opposite direction.

The dot product is positive when θ < 90°, zero when θ = 90°, and negative when θ > 90°. It captures how much the vectors "agree in direction."

Vector Projection — Decomposing u Along v

The projection of u onto v decomposes u into two perpendicular parts: one parallel to v and one perpendicular.

The scalar projection is the signed length of u's shadow onto v. Multiply by the unit vector of v to get the actual vector projection.

Try the example ⟨3,4⟩ · ⟨1,2⟩ to see the projection drawn as a dashed coral arrow on the diagram.