Vector Calculator

A vector is a mathematical object that has both a magnitude (length) and a direction. In 2D or 3D space, vectors are represented by their components along each axis (x, y, and z). They are widely used in physics, engineering, and computer graphics to represent quantities like velocity, force, and displacement.

To find the vector from point P(x₁, y₁, z₁) to point Q(x₂, y₂, z₂), subtract the coordinates of P from Q: the result is the vector [x₂−x₁, y₂−y₁, z₂−z₁]. Enter these component differences as Vector A or B in this calculator.

The norm (or magnitude) of a vector is its length, calculated as √(x² + y² + z²). To normalize a vector means to scale it so its length equals 1 (a unit vector). You divide each component by the norm: â = a / |a|. Select 'Normalize Vector A' in this calculator to get the unit vector automatically.

The dot product (scalar product) of two vectors A and B is calculated as A·B = AxBx + AyBy + AzBz. It returns a single number (scalar), not a vector. If the dot product is zero, the vectors are perpendicular. It is commonly used to find the angle between vectors and in projections.

The cross product of two 3D vectors A and B produces a new vector perpendicular to both. It is computed using a determinant formula: the result has components (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx). The cross product is useful in physics for calculating torque, angular momentum, and the normal to a plane.

The angle θ between two vectors A and B is found using the dot product formula: cos(θ) = (A·B) / (|A| × |B|), then θ = arccos(A·B / (|A||B|)). The result is in degrees. If the angle is 90°, the vectors are orthogonal; if it is 0° or 180°, they are parallel.

The projection of vector A onto vector B is the component of A that lies in the direction of B. It is calculated as proj = (A·B / |B|²) × B. This gives a vector pointing in the direction of B with magnitude equal to how much A extends along B.

The magnitude is the square root of the sum of squared components: √(3²+1²+4²+1²+5²) = √(9+1+16+1+25) = √52 ≈ 7.211. This calculator supports up to 3D vectors (x, y, z), which covers the most common physics and engineering use cases.