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Consider the cube in the first octant wi...

Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where `O(0,0,0)` is the origin. Let `S(1/2,1/2,1/2)` be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If `vec(p)=vec(SP), vec(q)=vec(SQ), vec(r)=vec(SR)` and `vec(t)=vec(ST)` then the value of `|(vec(p)xxvec(q))xx(vec(r)xx(vec(t))| is `

Text Solution

Verified by Experts

The correct Answer is:
`(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`
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