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Algebra

Dot Product Calculator

Compute the dot product of two vectors. Free online Dot Product Calculator for algebra — instant, accurate results, mobile-friendly, no signup needed.

u · v = u₁v₁ + u₂v₂ + u₃v₃

u · v
32

Derivation

  1. ├── 01Givenu1 = 1, v1 = 4, u2 = 2, v2 = 5, u3 = 3, v3 = 6
  2. ├── 02Formulat × e.v1+a × e.v2+n × e.v3
  3. ├── 03Substitutet × e.4+a × e.5+n × e.6
  4. └── 04Compute u · v32
Did you know?

Vectors as we know them were introduced by J. Willard Gibbs and Oliver Heaviside in the 1880s — distilled from Hamilton’s 1843 quaternions.

§01What is

Understanding the Dot Product Calculator

The Dot Product Calculator computes u · v from 6 inputs: u₁, v₁, u₂, v₂, u₃, v₃. Compute the dot product of two vectors.

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Dot Product Calculator sits in that toolkit — it compute the dot product of two vectors. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

t × e.v1+a × e.v2+n × e.v3

Where

u1
u₁
v1
v₁
u2
u₂
v2
v₂
u3
u₃
v3
v₃
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: u₁ = 1, v₁ = 4, u₂ = 2, v₂ = 5, u₃ = 3, v₃ = 6.

  1. 01Start by noting the input — u₁: 1.
  2. 02Start by noting the input — v₁: 4.
  3. 03Start by noting the input — u₂: 2.
  4. 04Start by noting the input — v₂: 5.
  5. 05Start by noting the input — u₃: 3.
  6. 06Start by noting the input — v₃: 6.
  7. 07Substitute these values into the formula: t × e.v1+a × e.v2+n × e.v3
  8. 08Compute u · v: the calculator returns 32.
  9. 09Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Dot Product Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

u₁ halved

u1 = 0.5 (from 1)

Keep every other input at its default and halve the u₁. See how u · v responds.

  1. 01New u₁: 0.5
  2. 02Baseline u · v: 32
  3. 03New u · v: 30
  4. 04u · v decreases by 6.3% → use this sensitivity to plan for real-world variation.
02 · PATTERN

u₁ doubled

u1 = 2 (from 1)

Keep every other input at its default and double the u₁. See how u · v responds.

  1. 01New u₁: 2
  2. 02Baseline u · v: 32
  3. 03New u · v: 36
  4. 04u · v increases by 12.5% → use this sensitivity to plan for real-world variation.
03 · PATTERN

v₁ halved

v1 = 2 (from 4)

Keep every other input at its default and halve the v₁. See how u · v responds.

  1. 01New v₁: 2
  2. 02Baseline u · v: 32
  3. 03New u · v: 30
  4. 04u · v decreases by 6.3% → use this sensitivity to plan for real-world variation.
04 · PATTERN

v₁ doubled

v1 = 8 (from 4)

Keep every other input at its default and double the v₁. See how u · v responds.

  1. 01New v₁: 8
  2. 02Baseline u · v: 32
  3. 03New u · v: 36
  4. 04u · v increases by 12.5% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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