Free online vector calculators for cross product, dot product, unit vector calculation, and vector operations. Calculate 2D and 3D vector properties with step-by-step solutions.
Vector mathematics is a fundamental branch of linear algebra that deals with quantities having both magnitude and direction. Our comprehensive vector calculator suite provides tools for performing essential vector operations including cross product, dot product, unit vector calculation, and more. Whether you're a student learning vector calculus, an engineer working with 3D modeling, or a physicist analyzing forces, our calculator offers accurate results with detailed explanations.
This vector toolset helps you calculate vector properties in both 2D and 3D space, convert between different vector representations, and understand the geometric relationships between vectors. Each operation includes step-by-step solutions to enhance your understanding of vector mathematics.
Enter vector components to calculate cross product
For vectors a = (1, 2, 3) and b = (4, 5, 6), the cross product a×b = (-3, 6, -3), which is perpendicular to both original vectors.
For vectors a = (2, 3) and b = (4, 1), the dot product a·b = 2×4 + 3×1 = 11, which relates to the angle between vectors.
For vector v = (3, 4), the unit vector is v/|v| = (3, 4)/5 = (3/5, 4/5), which has magnitude 1 but same direction as v.
Calculating torque: τ = r × F, where r is position vector and F is force vector. The cross product gives the rotational effect of the force.
Calculating forces, torques, and moments in mechanical systems
Analyzing electromagnetic fields, momentum, and motion in 3D space
3D modeling, lighting calculations, and normal vector computations
Vectors are quantities with both magnitude and direction (like velocity, force, displacement), while scalars have only magnitude (like speed, temperature, mass). Vectors are represented as arrows in space or as ordered lists of components.
The dot product of two vectors results in a scalar quantity. It's calculated as the sum of the products of corresponding components. Geometrically, a·b = |a||b|cos(θ), where θ is the angle between the vectors. The dot product indicates how much one vector extends in the direction of another.
The cross product of two vectors results in another vector that is perpendicular to both original vectors. It's calculated using the determinant method for 3D vectors. Geometrically, |a×b| = |a||b|sin(θ), representing the area of the parallelogram formed by the vectors.
A unit vector has a magnitude of 1 and indicates direction. Any vector can be converted to a unit vector by dividing by its magnitude. Unit vectors in the direction of coordinate axes are called standard unit vectors: i (1,0,0), j (0,1,0), k (0,0,1).
The dot product results in a scalar value and measures how much two vectors are aligned (parallel). The cross product results in a vector perpendicular to both input vectors and measures how much the vectors are perpendicular (orthogonal). Dot product is commutative, while cross product is anti-commutative.
For a 2D vector (x, y), the magnitude is √(x² + y²). For a 3D vector (x, y, z), the magnitude is √(x² + y² + z²). The magnitude represents the length of the vector.
Unit vectors are used when you need to represent direction without magnitude. They're essential in physics for representing directions of forces, in computer graphics for surface normals, and in navigation for heading directions.
The true cross product is only defined for 3D vectors, resulting in another 3D vector. However, for 2D vectors, you can compute a pseudoscalar cross product that gives the signed magnitude of the resulting perpendicular vector.
A negative dot product means the angle between the vectors is greater than 90° (obtuse). The vectors point in directions more opposite than perpendicular. A positive dot product means the angle is less than 90° (acute).
Yes, this calculator provides the foundational understanding needed for advanced mathematics including vector calculus, linear algebra, and differential geometry. The step-by-step solutions help reinforce concepts needed in higher-level math and physics courses.