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Linear Algebra
University
Linear Algebra (University Level)
Dot Product of Vectors
Calculate the dot product of two 3D vectors
Standard Formula
a · b = a₁b₁ + a₂b₂ + a₃b₃
Interactive Sandbox
Live Reactive
Adjust values in real-time and observe mathematical cause-and-effect
a1
0.120
a2
120
a3
120
b1
120
b2
120
b3
120
Live EquationFormula Proof
a · b = a₁b₁ + a₂b₂ + a₃b₃ = 32
Dynamic Formula Model
Calculated Solution
32
Direct computational model evaluating Dot Product of Vectors.
Variable Mechanics
Roles, constraints & impacts
6 Params
a1
a₁
a₁
ImpactPrimary formula parameter
ConstraintReal number
a2
a₂
a₂
ImpactPrimary formula parameter
ConstraintReal number
a3
a₃
a₃
ImpactPrimary formula parameter
ConstraintReal number
b1
b₁
b₁
ImpactPrimary formula parameter
ConstraintReal number
b2
b₂
b₂
ImpactPrimary formula parameter
ConstraintReal number
b3
b₃
b₃
ImpactPrimary formula parameter
ConstraintReal number
At a Glance
Gotchas, units & real-world use
Essential
Formula Nature & Mechanics
Direct computational model evaluating Dot Product of Vectors.
Common Mistakes & Traps
- Ensure all input measurements share consistent units before calculating.
- Check for denominator values approaching zero to prevent undefined division.
- Follow standard order of operations (PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract).
Dimensional Analysis & Units
Direct evaluation matching formula variable dimensions.
Real-World & Industry Application
Widely utilized across modern linear algebra applications, academic analysis, and engineering estimations.
Worked Example: Dot Product of Vectors
Given:
- a1 = 1
- a2 = 2
- a3 = 3
- b1 = 4
- b2 = 5
- b3 = 6
- a · b = (1)(4) + (2)(5) + (3)(6)
- = 4 + 10 + 18
- = 32
Answer: 32
Frequently Asked Questions
What is the Dot Product of Vectors formula?
a · b = a₁b₁ + a₂b₂ + a₃b₃. Calculate the dot product of two 3D vectors.
How do I calculate Dot Product of Vectors?
Enter a₁ (a1), a₂ (a2), a₃ (a3), b₁ (b1), b₂ (b2), b₃ (b3) into the calculator. It applies a · b = a₁b₁ + a₂b₂ + a₃b₃ and shows every step of the working.
Related Linear Algebra Calculators
2×2 Matrix Determinantdet(A) = ad - bcVector Magnitude|v| = √(x² + y² + z²)Eigenvalues of 2×2 Matrixλ = (a+d ± √((a+d)²-4(ad-bc)))/22×2 Matrix Multiplication[a b; c d] × [e f; g h] = [ae+bg af+bh; ce+dg cf+dh]2×2 Matrix InverseA⁻¹ = (1/det) × [d -b; -c a], det = ad - bc3×3 Matrix Determinantdet = a(ei - fh) - b(di - fg) + c(dh - eg)
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