$ jq -r -n -f vector.jq
v     is [1,1]
    w is [3,4]
v + w is [4,5]
v - w is [-2,-3]
  - v is [-1,-1]
w * 5 is [15,20]
w / 2 is [1.5,2]
v|topolar is [1.4142135623730951,0.7853981633974483]
w|topolar is [5,0.9272952180016122]
z = polar(1; pi/2) is [6.123233995736766e-17,1]
z|topolar is [1,1.5707963267948966]
z2 = polar(-2; pi/4) is [-1.4142135623730951,-1.414213562373095]
z2|topolar is [2,-2.356194490192345]
z2|topolar|polar is [-1.414213562373095,-1.4142135623730951]