Correct Answer - `(0.5)`
(0.5) Here P(1,0,0) Q(0,1,0) R (0,0,1) , T =(1,1,1) and
`S= ((1)/(2) ,(1)/(2),(1)/(2))`
Now ` vec(P) = vec(SP) =vec(OP) - vec(OS)`
`= ((1)/(2) hat(i) -(1)/(2) hat(j) -(1)/(2) hat(k)) = (1)/(2) (hat(i) - hat(j) - hat(k))`
` vec(q) = vec(SQ) = (1)/(2) (hat(i) + hat(j) - hat(k))`
`vec(r ) = vec(SR) = (1)/(2) (-hat(i) - hat(j) + hat(k))`
And `vec(t) = vec(ST) = (1)/(2) (hat(i) + hat(j) + hat(k))`
`vec(p) xx vec(q) = (1)/(4) |{:(hat(i) ,,hat(j) ,,hat(k) ),(1,,-1,,-1),(-1,,1,,-1):}| =(1)/(4) (2hat(i) + 2hat(j))`
` and vec(r ) xx vec(t ) = (1)/(4) |{:(hat(i) ,,hat(j) ,,hat(k)),(-1,,-1,,1),(1,,1,,1):}| =(1)/(4) (-2hat(i) + 2hat(j))`
Now `(vec(p) xx vec(q)) xx (vec(r ) xx vec(t)) = (1)/(16) |{:(hat(i) ,,hat(j),,hat(k) ),(2,,2,,2),(-2,,2,,0):}|`
`=(1)/(16) (8hat(k)) = (1)/(2) hat(k)`
`:. (p xx q) xx (vec(r ) xx vec(t)) = |(1)/(2) hat(k)|= (1)/(2) = 0.5`