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A tetrahedron has vertices P(1,2,1),Q(2,...

A tetrahedron has vertices `P(1,2,1),Q(2,1,3),R(-1,1,2) and O(0,0,0)`. The angle beween the faces OPQ and PQR is :

A

`cos^(-1)((9)/(35))`

B

`cos^(-1)((17)/(31))`

C

`cos^(-1)((7)/(31))`

D

`cos^(-1)((19)/(35))`.

Text Solution

Verified by Experts

The correct Answer is:
D
Vector perpendicular to face OPQ
`vec(n_1)=vec(OP)xxvec(OQ)=|(i,j,k),(1,2,1),(2,1,3)|=5i-j-3k`
Vector perpendicular to face PQR
`vec(n_2)=vec(PQ)xxvec(PR)=|(i,j,k),(-2,-1,1),(1,-1,2)|=-i+5j+3k`
`therefore ` Angle between two faces is given by
`cos theta=|(-5-5-9)/(sqrt(35)sqrt(35))|=(19)/(35)`
`theta=cos^(-1)((19)/(35))`.
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