🔍 /

3D/2D Vector Operations Solver

Calculate dot product, cross product, magnitude, and angles for 2D and 3D vectors. Includes visual canvas representation of vector addition.

User Rating 0.0 ★★★★★
Total Usage 1 times
Category Math & Algebra

Vector A

i
j
k

Vector B

i
j
k
A + B:
A − B:
Dot Product:
Cross Product:
Angle:
|A|: |B|:
Is this tool helpful?

Your feedback helps us improve.

★ ★ ★ ★ ★

About

Vectors are fundamental to physics and engineering, representing quantities with both magnitude and direction. While manual calculation of the cross product for 3D vectors involves complex determinant matrices, errors are common. This tool automates these linear algebra operations. It supports both R2 (2D plane) and R3 (3D space) systems.

Visualizing the resulting vector from a Cross Product (orthogonal to the plane of the two input vectors) is notoriously difficult for students. This solver provides the numeric components along with magnitude and angle calculations, essential for mechanics and electromagnetism problems.

vector calculator cross product dot product linear algebra physics vectors

Formulas

Cross Product Determinant:

a × b = ijkaxayazbxbybz

Resulting Components:

cx = aybz − azby

cy = azbx − axbz

cz = axby − aybx

Reference Data

OperationFormulaResult Type
Magnitude|v| = √(x2 + y2 + z2)Scalar
Dot Producta ⋅ b = x1x2 + y1y2...Scalar
Cross Producta × bVector
Angle (θ)cos-1( (a ⋅ b) / (|a| |b|) )Degrees/Radians
Unit Vectoru = v / |v|Vector
Projectionprojba = ( (a ⋅ b) / |b|2 ) bVector
Distance|a - b|Scalar
Area (Parallelogram)|a × b|Scalar

Frequently Asked Questions

The Cross Product results in a vector that is perpendicular (orthogonal) to the plane defined by the two input vectors. Its magnitude is equal to the area of the parallelogram spanned by the vectors.
Yes. For 2D operations, simply leave the Z-component as 0. The Dot Product and Magnitude will calculate correctly. The Cross Product will result in a vector pointing purely in the Z direction (0, 0, k).
The Dot Product multiplies the magnitudes of two vectors and the cosine of the angle between them. It represents how much one vector goes in the direction of the other, resulting in a single number (energy, work) rather than a direction.