Vector Calculator

Frequently Asked Questions

How do you find a vector's magnitude?

Magnitude = √(x² + y² [+ z²]) - the Pythagorean length. The unit vector is each component divided by the magnitude.

What is the difference between dot and cross product?

The dot product is a scalar (a·b = |a||b|cosθ), measuring alignment. The cross product (3D) is a vector perpendicular to both, with magnitude |a||b|sinθ.

How do I find the angle between two vectors?

θ = arccos((a·b)/(|a||b|)). A dot product of zero means the vectors are perpendicular.

What is the cross product used for?

Surface normals in graphics, torque and angular momentum in physics, and testing whether 3D vectors are parallel (cross product is the zero vector).

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Estimates for informational purposes only.

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How This Calculator Works

A vector is an ordered tuple with magnitude and direction. Magnitude uses the Pythagorean theorem extended to n dimensions: |v| = √(v_x² + v_y² + v_z²). Addition and subtraction are component-wise (a + b)_i = a_i + b_i, the geometric tip-to-tail rule. Scalar multiplication scales every component, stretching the vector while keeping (or reversing) its line. The dot product a·b = Σa_ib_i = |a||b|cosθ measures alignment and yields the angle between vectors: θ = arccos(a·b ÷ (|a||b|)). The cross product (3D only) a×b is a vector perpendicular to both, with magnitude |a||b|sinθ equal to the area of the parallelogram they span and direction given by the right-hand rule. The unit vector â = a ÷ |a| keeps direction with length 1. This tool computes all of these for two 2D or 3D vectors at once.

Worked Example: a = (1, 2, 3), b = (4, 5, 6)

|a| = √(1 + 4 + 9) = √14 ≈ 3.7417. a + b = (5, 7, 9); a − b = (−3, −3, −3). Dot product = 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32. |b| = √(16+25+36) = √77 ≈ 8.7750, so cosθ = 32 ÷ (3.7417 × 8.7750) ≈ 0.9746, giving θ ≈ 12.93° - the vectors are nearly parallel. Cross product a×b = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (12−15, 12−6, 5−8) = (−3, 6, −3); its magnitude √54 ≈ 7.348 is the area of the parallelogram spanned by a and b. Unit vector of a ≈ (0.2673, 0.5345, 0.8018).

Where Vectors Are Used

Physics: force, velocity, acceleration, momentum - torque is r × F, work is F · d.

Computer graphics & games: surface normals (cross product), lighting (dot product), camera and movement vectors.

Machine learning: feature vectors; cosine similarity is the normalized dot product used in search and embeddings.

Engineering & navigation: resolving forces into components and combining displacement headings.

Common Mistakes & Edge Cases

Treating the dot product (a scalar) and cross product (a vector) as interchangeable - they answer different questions: alignment vs perpendicularity.

Cross product is 3D-only and non-commutative: a×b = −(b×a). Order matters; swapping flips the normal.

Floating-point can push cosθ slightly past ±1, making arccos return NaN - this tool clamps to [−1, 1].

The zero vector has no defined direction, so it has no unit vector - reported as undefined here.

Adding vectors of different dimensions is undefined; switch the 2D/3D toggle so both match.

Related Concepts & Limits

The dot product generalizes to the inner product on any vector space and underpins projections: the scalar projection of a onto b is a·b ÷ |b|. The cross product is special to 3D (and 7D); in n dimensions the analogous object is the wedge product of exterior algebra. Orthogonal vectors have a zero dot product; parallel vectors have a zero cross product - useful tests for perpendicularity and collinearity. Cosine similarity (dot product of unit vectors) ignores magnitude and compares pure direction, which is why it dominates text and embedding search. Computations here use double precision (≈ 15 significant digits); angles are returned in degrees for readability and clamped to avoid arccos domain errors.

Vector Calculator

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