a · b = ∑ aibi = |a||b|cosθ
Vector Calculator with Steps
Dot · cross · magnitude · projection · 13 ops · 2D/3D graph · NumPy export
13 Operations
2D & 3D
Step-by-Step
Plotly Graph
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Enter vector components above…
Result
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Enter vectors and click Calculate
Supports 13 vector operations including dot product, cross product, projection, and more.
Interactive Graph
Calculate vectors to see the graph.
Python (NumPy)
a × b ⊥ a, b |a×b| = area
proja(b) = [(a·b)/(a·a)] a
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About this vector calculator
Compute dot product, cross product, magnitude, unit vector,
angle, projection, rejection, parallelogram area,
triple scalar product, and linear independence for 2D and 3D vectors with full LaTeX steps.
Use Math AI for tutoring or to solve vector problems in chat via ```vector``` blocks (Solve / Steps chips).
| Operation | Formula | Returns | Dim |
|---|---|---|---|
| Addition | a + b = (ai+bi) | Vector | 2D/3D |
| Dot Product | ∑ aibi | Scalar | 2D/3D |
| Cross Product | det[î ĵ k̂; a; b] | Vector | 3D |
| Projection | (a·b/a·a)·a | Vector | 2D/3D |
| Triple Scalar | a · (b × c) | Scalar | 3D |
FAQ
What is the dot product of two vectors?
The sum of products of corresponding components. Zero means orthogonal. Also equals |a||b|cos(θ).
How do you compute the cross product?
Use the 3×3 determinant with î, ĵ, k̂ in the first row. Result is perpendicular to both inputs (3D only).
What is vector projection?
proja(b) = (a·b)/(a·a) × a gives the component of b along a; rejection is the orthogonal remainder.
How do you find the angle between two vectors?
θ = arccos((a·b)/(|a||b|)). The calculator shows radians and degrees.