Vector Calculator: Addition, Dot and Cross Product

What is a vector?

A vector is a quantity with both size and direction: force, velocity and displacement are vectors. A vector is given either by its components (A = (3, 4)) or by its magnitude and angle (5 units at 53.13°). This tool does four jobs with two and three dimensional vectors: addition and resultants, the dot product and angle, the cross product and the triple product.

Addition and resultants

Vectors are added component by component. A vector given by magnitude and angle is first resolved into components: Ax = |A| · cos θ and Ay = |A| · sin θ. Two perpendicular forces of 3 and 4 units have the resultant (3, 4). Its magnitude is √(9 + 16) = 5 and its direction is arctan(4/3) = 53.13°. The tool adds up to six vectors. To subtract a vector, set its sign to minus.

Dot product and angle

The dot product is a number: A · B = Ax·Bx + Ay·By + Az·Bz. Because the same product equals |A| · |B| · cos θ, it gives the angle between two vectors. For A = (2, 2, −1) and B = (5, −3, 2) you get A · B = 2, |A| = 3, |B| = √38 and θ = 83.79°. If the dot product is zero the vectors are perpendicular.

Cross product

The cross product is a new vector perpendicular to both. Its magnitude equals the area of the parallelogram spanned by the two vectors. For A = (1, 3, 4) and B = (2, 7, −5) the result is A × B = (−43, 13, 1). Swapping the order flips the sign: B × A = −(A × B).

Triple product

The number A · (B × C) is the signed volume of the parallelepiped spanned by three vectors. If it is zero the three vectors lie in one plane.

Limits

Enter all vectors in the same unit. The tool does not convert units. The plot is drawn in two dimensions only. The work is done in decimals and exact surd forms are not produced.