Vector Dot Product Calculator
Enter two 3-D vectors to compute the dot product a·b = ax·bx+ay·by+az·bz, their magnitudes |a|, |b| and the angle θ between them. a=(1,2,3), b=(4,5,6) → a·b=32, θ≈12.93°.
Input Data
Results
At a glance:The dot product (scalar product) of two 3-D vectors a and b is defined algebraically as a·b = a_x·b_x + a_y·b_y + a_z·b_z, and geometrically as a·b = |a|·|b|·cosθ, where |a|, |b| are the vector magnitudes and θ is the angle between them. It is a scalar (single number), not a vector — hence 'scalar product'. Key properties: (1) a·b = 0 means the vectors are perpendicular (orthogonal); (2) a·b > 0 means the angle is acute (< 90°), a·b < 0 means obtuse (> 90°); (3) the magnitude of the dot product is maximal when the vectors are parallel (θ=0, a·b=|a||b|); (4) a·a = |a|². Example (default): a=(1,2,3), b=(4,5,6) → a·b=1·4+2·5+3·6=4+10+18=32; |a|=√(1+4+9)=√14≈3.742, |b|=√(16+25+36)=√77≈8.775; cosθ=32/(3.742×8.775)=0.974, θ≈12.93°. Applications: (1) work done by a force W=F·d (force dotted with displacement); (2) vector projection proj_b(a)=(a·b/|b|²)b; (3) diffuse lighting in computer graphics (normal·light); (4) testing orthogonality; (5) the law of cosines. Notes: (1) the dot product is commutative a·b=b·a; (2) units combine from both vectors; (3) if magnitudes are zero the angle is undefined.
Formula
Algebraic: a·b = a_x·b_x + a_y·b_y + a_z·b_z
Geometric: a·b = |a|·|b|·cosθ
Magnitude: |a| = √(a_x²+a_y²+a_z²)
Angle: θ = arccos(a·b / (|a|·|b|))
Perpendicular ⇒ a·b = 0
$$\mathbf{a}\cdot\mathbf{b} = a_x b_x + a_y b_y + a_z b_z = |\mathbf{a}||\mathbf{b}|\cos\theta$$How to Use
- Enter the x, y, z components of vector a and vector b.
- The tool shows a·b, |a|, |b| and the angle θ between them.
- a·b=0 means the vectors are perpendicular.
Case Studies
Work done by a force
Force F=(10,0,0) N, displacement d=(3,4,0) m.
W = F·d = 10×3 + 0×4 + 0×0 = 30 J.
Only the component of force along displacement does work; the perpendicular part contributes nothing.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.