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Free vector multiplication calculator. Calculate dot product, cross product, and scalar multiplication of vectors with step-by-step solutions. Supports 2D and 3D vectors for linear algebra and physics applications. Our calculator handles all major vector multiplication operations with detailed explanations.
Last updated: February 2, 2026
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Angle
10.3048°
Magnitude
11.0000
Formula:
A · B = 3×1 + 4×2 = 11.0000
Analysis:
The vectors point in generally the same direction (acute angle).
The dot product of vectors A = ⟨3, 4⟩ and B = ⟨1, 2⟩ is 11.0000. The angle between the vectors is 10.3048°.
Formula
A · B = Σ(Ai·Bi)
Results in a scalar value, measures vector alignment
Formula
A × B = (Ay·Bz - Az·By, ...)
Results in a vector, measures perpendicular area
Formula
k × A = (k·Ax, k·Ay, k·Az)
Scales vector magnitude while preserving direction
Alternative name
Scalar Product
Another name for the dot product operation
Result type
Perpendicular Vector
Produces a vector perpendicular to both input vectors
Operations
All Multiplication Types
Supports dot product, cross product, and scalar multiplication
Dot product of A = ⟨3, 4⟩ and B = ⟨1, 2⟩:
Dot Product
A · B = 11
Angle
θ ≈ 7.13°
Our vector multiplication calculator performs dot product, cross product, and scalar multiplication operations using fundamental linear algebra formulas. The calculations apply vector algebra principles to compute scalar and vector results based on component-wise operations and geometric interpretations.
A · B = Ax·Bx + Ay·By + Az·Bz (dot product)A × B = (Ay·Bz - Az·By, Az·Bx - Ax·Bz, Ax·By - Ay·Bx) (cross product, 3D)k × A = (k·Ax, k·Ay, k·Az) (scalar multiplication)A · B = |A| × |B| × cos(θ) (geometric dot product)The dot product gives a scalar result indicating vector alignment. The cross product gives a vector perpendicular to both inputs (3D only). Scalar multiplication scales the vector's magnitude.
Shows dot product, cross product, and scalar multiplication operations
Vector multiplication operations are fundamental to linear algebra and have geometric interpretations. The dot product measures projection and angle relationships, the cross product finds perpendicular vectors and areas, and scalar multiplication performs uniform scaling. These operations form the basis for advanced vector calculus, physics applications, and computer graphics.
Need help with other vector calculations? Check out our vector addition calculator and cross product calculator.
Get Custom Calculator for Your PlatformResult: Dot Product = 11, Angle ≈ 7.13°
The vectors point in generally the same direction with a small angle between them.
A = ⟨1, 0, 0⟩, B = ⟨0, 1, 0⟩
A × B = ⟨0, 0, 1⟩
Perpendicular to both input vectors
A = ⟨3, 4⟩, k = 2
2 × A = ⟨6, 8⟩
Magnitude doubled, direction preserved
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