Calculator Inputs
Calculation History
| Time | Input | Dot product | Angle | Relationship |
|---|---|---|---|---|
| No saved calculations yet. | ||||
Formula Used
Component form: A · B = a₁b₁ + a₂b₂ + ... + aₙbₙ.
Magnitude-angle form: A · B = |A||B|cos(θ).
Angle: θ = cos⁻¹((A · B)/(|A||B|)).
The dot product combines matching vector components. It returns one scalar value. This value measures directional agreement between both vectors clearly.
How To Use
Select components or the magnitude-angle method. Enter valid values in every visible field. Then press calculate to view complete working steps.
Use fractions, decimals, or scientific notation directly. Match both dimensions when using vector components. Projection results require a nonzero reference vector always.
Choose precision before submitting your calculation. Export results using CSV, PDF, or printing. Your recent calculations stay inside this browser only.
Example Data
| Example | Vector A | Vector B | Dot product | Relationship |
|---|---|---|---|---|
| Basic 2D | (2, 3) | (4, 5) | 23 | Acute |
| Three dimensions | (1, 2, 3) | (4, -5, 6) | 12 | Acute |
| Orthogonal | (1, 0) | (0, 1) | 0 | Right angle |
| Parallel | (1, 2) | (2, 4) | 10 | Same direction |
| Fractions | (1/2, 3/4) | (2, 4) | 4 | Acute |
| Physics work | (10, 0) | (3, 4) | 30 | Positive work |
Understanding Dot Products
A positive result means vectors generally agree. A negative result means they oppose. Zero indicates perpendicular directions under nonzero magnitudes exactly.
Scalar projection measures signed length along another vector. Vector projection includes direction and component coordinates. Rejection captures the perpendicular portion remaining after projection.
Dot products support geometry and physical work. Graphics uses them for lighting calculations. Machine learning uses them for similarity scoring frequently.
Frequently Asked Questions
What is a dot product?
A dot product multiplies matching vector components. Those products are then added together. The final answer is always one scalar value.
Can vectors have more than three dimensions?
Yes, dot products support any matching dimension. This calculator accepts dimensions from two through twenty. Every component pair contributes one multiplication to totals.
When are two vectors orthogonal?
Nonzero vectors are orthogonal when their dot product equals zero. Their included angle then measures ninety degrees. Small rounding differences may require a practical tolerance.
What happens with a zero vector?
The dot product still equals zero. However, its direction and angle remain undefined. Unit vectors and reference projections also become undefined.
How does vector projection differ from scalar projection?
Scalar projection gives a signed length. Vector projection gives coordinates with direction. Both describe alignment along the selected reference vector.
Can I enter fractions?
Yes, enter fractions using numerator and denominator notation. Denominators must never equal zero. Exact dot products appear when every input remains rational.
What does a negative dot product mean?
A negative result indicates general directional opposition. The included angle exceeds ninety degrees. In physics, this can represent negative mechanical work.
How is the angle calculated?
The calculator divides the dot product appropriately. It then applies the inverse cosine function. Both vector magnitudes must be greater than zero.
How does the dot product differ from a cross product?
A dot product returns one scalar. A cross product returns a perpendicular vector. Cross products normally apply only within three-dimensional vector spaces.