# Dot Product Calculator

Calculate the dot product of two 2D or 3D vectors. Find the angle between vectors, check orthogonality, and compute magnitudes.

## What this calculates

Compute the dot product (scalar product) of two vectors, find the angle between them, and determine whether they are orthogonal. Enter the components of each vector and get instant results.

## Inputs

- **V1 x** — X component of vector 1.
- **V1 y** — Y component of vector 1.
- **V1 z** — Z component of vector 1 (set to 0 for 2D).
- **V2 x** — X component of vector 2.
- **V2 y** — Y component of vector 2.
- **V2 z** — Z component of vector 2 (set to 0 for 2D).

## Outputs

- **Dot Product** — The scalar dot product A · B.
- **Angle Between Vectors** — The angle between the two vectors in degrees.
- **Orthogonal?** — formatted as text — Whether the two vectors are perpendicular (dot product = 0).
- **Magnitude of V1** — The magnitude (length) of vector 1.
- **Magnitude of V2** — The magnitude (length) of vector 2.

## Details

The dot product is one of the most important operations in vector algebra. For two vectors A = (a1, a2, a3) and B = (b1, b2, b3), the dot product is A · B = a1b1 + a2b2 + a3*b3. The result is a scalar, not a vector.

Geometrically, the dot product relates to the angle between the vectors: A · B = |A| |B| cos(theta). This means you can find the angle using theta = arccos(A · B / (|A| |B|)). When two vectors are perpendicular (orthogonal), their dot product is zero because cos(90°) = 0.

The dot product has many applications: computing projections, testing perpendicularity, calculating work done by a force along a displacement, determining lighting in computer graphics (Lambert's cosine law), and finding the component of one vector along another direction.

## Frequently Asked Questions

**Q: What is the dot product?**

A: The dot product of two vectors A and B is the sum of the products of their corresponding components: A · B = a1b1 + a2b2 + a3*b3. It produces a scalar value, not a vector, and measures how much one vector extends in the direction of another.

**Q: How do I find the angle between two vectors?**

A: Use the formula theta = arccos(A · B / (|A| × |B|)), where A · B is the dot product and |A|, |B| are the magnitudes. The angle ranges from 0° (parallel, same direction) to 180° (parallel, opposite direction), with 90° meaning perpendicular.

**Q: What does it mean when the dot product is zero?**

A: A dot product of zero means the two vectors are orthogonal (perpendicular). The angle between them is exactly 90°. This is a key test in geometry, physics, and linear algebra for checking if vectors are independent directions.

**Q: What is the difference between dot product and cross product?**

A: The dot product yields a scalar and measures how parallel two vectors are. The cross product yields a vector perpendicular to both input vectors and measures how perpendicular they are. Dot product uses cosine of the angle; cross product uses sine.

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Source: https://vastcalc.com/calculators/math/dot-product
Category: Math
Last updated: 2026-04-21