Let a , b , c be non-coplanar and define α = a + 2 b + 3 c , β = a + 3 b + 4 c , γ = a + b + c , and λ = 21 a + 41 b + 54 c . If λ = x 1 α + x 2 β + x 3 γ , which statements are true?

Question

Let a , b , c be non-coplanar and define α = a + 2 b + 3 c , β ​ = a + 3 b + 4 c , γ ​ = a + b + c , and λ = 21 a + 41 b + 54 c . If λ = x 1 ​ α + x 2 ​ β ​ + x 3 ​ γ ​ , which statements are true?

TestHub · Accepted Answer

Comparing coefficients gives x 1 ​ + x 2 ​ + x 3 ​ = 21 , 2 x 1 ​ + 3 x 2 ​ + x 3 ​ = 41 , and 3 x 1 ​ + 4 x 2 ​ + x 3 ​ = 54 . Solving gives ( x 1 ​ , x 2 ​ , x 3 ​ ) = ( 6 , 7 , 8 ) .

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