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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)
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B
cos1(1935)
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C
cos1(1731)
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D
cos1(731)
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Solution

The correct option is B $$\cos ^ { - 1 } \left( \dfrac { 19 } { 35 } \right)$$
$$\vec { \mathrm { OP } } \times \vec { \mathrm { OQ } } = ( \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } ) \times ( 2 \hat { \mathrm { i } } + \hat { \mathrm { j } } + 3 \hat { \mathrm { k } } )$$
$$5 \hat { \mathrm { i } } - \hat { \mathrm { j } } - 3 \hat { \mathrm { k } }$$
$$\vec { \mathrm { PQ } } \times \vec { \mathrm { PR } } = ( \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \hat { \mathrm { k } } ) \times ( - 2 \hat { \mathrm { i } } - \hat { \mathrm { j } } + \hat { \mathrm { k } } )$$
$$\hat { \mathrm { i } } - 5 \hat { \mathrm { j } } - 3 \hat { \mathrm { k } }$$
$$\cos \theta = \dfrac { 5 + 5 + 9 } { ( \sqrt { 25 + 9 + 1 } ) ^ { 2 } } = \dfrac { 19 } { 35 }$$ 
1142920_1331535_ans_9e1c4f521f33442e9e1b890b9bcb4c55.png

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