Let three coterminous edges of a tetrahedron are a¯,b¯ and c¯ such that |a¯|$$=2$$,|b¯|$$=3$$,|c¯|$$=4$$, angle between a¯ and b¯ is π$$3$$,b¯ and c¯ is π$$4$$,c¯ and a¯ is π$$6$$. If area of the base is $$2$$ square units and height of the tetrahedron be $$h=lm-n$$ where l,m,n are positive integers and l is odd prime number then $$l+m-n$$ is

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Let three  coterminous edges of a tetrahedron are a¯,b¯ and c¯ such that |a¯|$$=2$$,|b¯|$$=3$$,|c¯|$$=4$$, angle between a¯ and b¯ is π$$3$$,b¯ and c¯ is π$$4$$,c¯ and a¯ is π$$6$$. If  area of the base is $$2$$ square units and height of the tetrahedron be $$h=lm-n$$ where l,m,n are positive integers and l is odd prime number then $$l+m-n$$ is

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a¯.a¯$$=4$$,   b¯.b¯$$=9$$,   c¯.c¯$$=16a$$¯⋅b¯$$=3b$$¯⋅c¯$$=62c$$¯⋅a¯$$=43$$[a¯b¯c¯]$$2=a$$¯⋅a¯a¯.b¯a¯⋅c¯b¯⋅a¯b¯⋅b¯b¯⋅c¯c¯⋅a¯c¯⋅b¯c¯⋅c¯$$=144(6$$−$$2)$$ ⇒a¯,b¯,c¯$$=126-2$$ Volume of the tetrahedron $$=16$$∥[abc]∣$$=26-2$$⇒$$13$$× base area × height $$=26-2$$⇒$$13$$×$$2$$×$$h=26-2$$⇒$$h=36-2l=3$$,$$m=6$$,$$n=2$$

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