Calculator for Vectors in ℜ³

A vector is a one-dimensional matrix, it has a length and a direction and is often displayed as an arrow. Here the two vectors are denoted as a⃗⃗ and bb⃗, the result, if it is a vector, as cc⃗. The length of a vector is always positive and the direction is an angle. In physics, forces are often described by vectors.
This calculator deals with vectors in three-dimensional space. Vectors can be added (+), subtracted (-), multiplied with a number (*), scalar product (•) and cross product (x) can be calculated. Further, the absolute values of the single vectors (|1| and |2|) and the angle between them can be computed as radian (∠rad), multiples of pi (∠π) or in degrees (∠°).

Multiplying a number by a vector is equivalent to multiplying each of its elements by that number.
When adding two vectors c⃗=a⃗+b⃗, the corresponding elements are added, i.e. c₁=a₁+b₁, c₂=a₂+b₂ und c₃=a₃+b₃.
The scalar product of two vectors a⃗•b⃗⃗ results in a number that is calculated using the formula a1*b1+a2*b2+a3*b3. If both vectors are parallel, it is equal to the product of the lengths of these two, if the vectors are perpendicular to each other, then it is zero, otherwise its absolute value lies in between.
The cross product a⃗xb⃗⃗ creates a new vector c⃗ with c1=a2*b3-a3*b2, c2=a3*b1-a1*b3 and c3=a1*b2-a2*b1. This vector is perpendicular to the other two, so all three vectors span the space ℜ³ if a⃗⃗ and b⃗⃗ were not parallel. The length of c⃗⃗ corresponds to the area of ​​the parallelogram between a⃗⃗ and b⃗⃗⃗.
The absolute value |a⃗| or |b⃗| represents the length of the corresponding vector, which is calculated using the formula |a⃗|=√a₁²+a₂²+a₃². The other vector and the multiplier are irrelevant in this calculation.
The multiplier is also irrelevant for the angle. The three ways of specifying an angle do not change the size of the angle.