Calculator
Example Data Table
| Vector A | Vector B | Dot Product | Cosine | Angle |
|---|---|---|---|---|
| [3, -2, 5] | [4, 1, -3] | -5 | -0.1591 | 99.15 degrees |
| [2, 6] | [5, 1] | 16 | 0.6202 | 51.66 degrees |
| [1, 0, 0] | [0, 1, 0] | 0 | 0 | 90 degrees |
Formula Used
Dot product: A · B = a₁b₁ + a₂b₂ + ... + aₙbₙ
Magnitude: |A| = √(a₁² + a₂² + ... + aₙ²)
Angle: cos θ = (A · B) / (|A||B|)
Scalar projection of A on B: compBA = (A · B) / |B|
Vector projection of A on B: projBA = ((A · B) / |B|²)B
How to Use This Calculator
Enter Vector A and Vector B with matching dimensions. Use commas, spaces, semicolons, or line breaks between values.
Select decimal places and a zero tolerance. The tolerance helps classify nearly perpendicular vectors.
Choose the angle and projection outputs. Press Calculate to view the result above the form.
Use Download CSV or Download PDF to save the same calculation in a file.
Dot Product Study Guide
The dot product is a compact way to compare two vectors. It multiplies matching components, then adds those products. The final value shows how strongly the vectors point in the same direction. A positive value means the angle is acute. A negative value means the angle is obtuse. A zero value means the vectors are perpendicular, when both vectors are nonzero.
Why the Result Matters
This calculator supports classwork, engineering notes, and analytic geometry. You can enter vectors with any equal dimension. Two dimensional and three dimensional vectors are common. Higher dimensions work the same way. The result can help measure similarity, work, projection, alignment, and direction. It also gives component products, magnitudes, cosine similarity, and angle values.
Angle and Projection Meaning
The dot product connects algebra with geometry. Algebra uses matching component products. Geometry uses magnitudes and the cosine of the included angle. Because both views describe the same value, you can solve for the angle when both vectors have length. Projection is also useful. It shows how much one vector falls along another vector. A longer projection means stronger alignment.
Good Input Practice
Enter each vector in the same order. Separate values with commas, spaces, semicolons, or line breaks. Keep both vectors the same length. Check signs carefully, because one negative component can change the result. Use more decimal places when vectors contain measured data. Use a tolerance when deciding whether a value is close enough to zero.
Reading the Output
Start with the dot product. Then review magnitudes and the cosine value. If the cosine is near one, the vectors point in almost the same direction. If it is near negative one, they point in nearly opposite directions. If it is near zero, they are almost perpendicular. Use the projection vector when you need the part of one vector that lies along the other. Export the report when you need a record for homework, worksheets, or project documentation.
Accuracy Tips
Small rounding changes can affect angle results. Keep original data when possible. Compare exact dot products before rounded values. For zero vectors, angle and projection are not defined. This page reports those cases clearly, so your interpretation stays consistent during final written review.
FAQs
What is a dot product?
A dot product is the sum of matching component products from two vectors. It returns one number, not another vector.
Can I enter three dimensional vectors?
Yes. Enter three values for Vector A and three values for Vector B. The calculator also accepts higher dimensions.
Why must both vectors have equal length?
Each component in one vector must multiply a matching component in the other vector. Unequal dimensions break that pairing.
What does a zero dot product mean?
For nonzero vectors, a zero dot product means the vectors are perpendicular. The calculator also checks your chosen tolerance.
What is cosine similarity?
Cosine similarity is the dot product divided by both vector magnitudes. It measures directional similarity between vectors.
Why is the angle sometimes undefined?
The angle is undefined when either vector has zero magnitude. A zero vector has no clear direction for angle comparison.
What is vector projection?
Vector projection gives the part of one vector that lies along another vector. It is useful for components and alignment.
Can I save the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report.