Vectors `a ,ba n dc` are of the same length and when taken pair-wise they form equal angles. If ` vec a= hat i+ hat ja n d vec b= hat j+ hat k ,` then find vector ` vec cdot`

Explanation

`|veca+vecb+vecc+vecd|^(2)=sum|veca|^(2)+2sumveca.vecb=4+2vecd.(veca+vecb+vecc) " " (veca,vecb,vecc "are mutually perpendicular")`
`Let vecd=lambdaveca+muvecb+v vecc. Then vecd.veca=vecd.vecb=vecd.vecc=costheta`Therefore,
`lambda=mu=v=costheta`
` lambda^(2)+mu^(2)+v^(2)=1Rightarrow3cos^(2)theta=1costheta=1/sqrt3`
`|veca+vecb+vecc+vecd|^(2)=4+-(2.3)/sqrt3=4+-2sqrt3`