Question

# A tetrahedron has vertices  $$\mathrm { P } ( 1,2,1 ),\mathrm { Q } ( 2,1,3 ) , \mathrm { R } ( - 1,1,2 )$$  and  $$\mathrm { O } ( 0,0,0 ) .$$  The angle between the faces  $$OPQ$$  and  $$PQR$$  is :

A
cos1(935)
B
cos1(1935)
C
cos1(1731)
D
cos1(731)

Solution

## The correct option is B $$\cos ^ { - 1 } \left( \dfrac { 19 } { 35 } \right)$$$$\vec { \mathrm { OP } } \times \vec { \mathrm { OQ } } = ( \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } ) \times ( 2 \hat { \mathrm { i } } + \hat { \mathrm { j } } + 3 \hat { \mathrm { k } } )$$$$5 \hat { \mathrm { i } } - \hat { \mathrm { j } } - 3 \hat { \mathrm { k } }$$$$\vec { \mathrm { PQ } } \times \vec { \mathrm { PR } } = ( \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \hat { \mathrm { k } } ) \times ( - 2 \hat { \mathrm { i } } - \hat { \mathrm { j } } + \hat { \mathrm { k } } )$$$$\hat { \mathrm { i } } - 5 \hat { \mathrm { j } } - 3 \hat { \mathrm { k } }$$$$\cos \theta = \dfrac { 5 + 5 + 9 } { ( \sqrt { 25 + 9 + 1 } ) ^ { 2 } } = \dfrac { 19 } { 35 }$$ Maths

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