Enter up to three vectors and see them as glowing arrows in 3D. Instantly get the dot product, cross product, angle between them, projection, parallelogram area, scalar triple product and parallelepiped volume, magnitudes, unit vectors and the plane through three points — with the working shown step by step.
Dot product a · b = aₓbₓ + a_yb_y + a_zb_z = |a||b| cos θ. It measures how much two vectors point the same way: positive when the angle is acute, zero when perpendicular, negative when obtuse. The projection of a onto b is (a·b / |b|²) b — the shadow a casts on b's line.
Cross product a × b = (a_yb_z − a_zb_y, a_zbₓ − aₓb_z, aₓb_y − a_ybₓ) is perpendicular to both, follows the right-hand rule and has length |a||b| sin θ — the area of the parallelogram they span. Half of that is the triangle's area. It is zero when the vectors are parallel.
Scalar triple product a · (b × c) is the signed volume of the parallelepiped built on the three vectors (a sixth of it is the tetrahedron's volume). Zero means the three are coplanar; the sign says whether (a, b, c) is right-handed.
Plane through three points A, B, C: the normal is n = (B − A) × (C − A) and the equation is n · (r − A) = 0, written ax + by + cz = d. The distance from the origin to the plane is |d| / |n|.
Worked example: a = (1, 0, 0) and b = (0, 1, 0) give a·b = 0 (perpendicular), a × b = (0, 0, 1) = k̂ and θ = 90° — the right-hand rule in its simplest form.