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in Mathematics by (58.2k points)

If vector{a, b, c} are three non-coplanar vectors, then find the linear relation between the vectors a+ 3b+ 4c, a - 2b+ 3c, a + 5b - 2c, 6a+14b+ 4c

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Suppose vector{x(a + 3b + 4c) + y(a - 2b + 3c) + z(a + 5b - 2c) = 6a + 14b + 4c}

=> (x + y + z)a + (3x – 2y + 5z)b + (4x + 3y – 2z)c = 6a + 14b + 4c

=> x + y + z = 6  (1)

=> 3x – 2y + 5z = 14  (2)

=> 4x + 3y – 2z = 4  (3)

2(1) + (2) => 5x + 7z = 26 => (4)

3(1) – (3) => –x + 5z = 14 => (5)

5(5) + (4) => 32z = 96 => z = 3 

(4) => 5x + 7(3) = 26 => 5x = 5 => x = 1; (1)

=> 1 + y + 3 = 6 => y = 2

vector{1(a + 3b + 4c) + 2(a - 2b + 3c) + 3(a + 5b - 2c) - 1(6a + 14b +  4c) = 0}

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