Given that the vectors $¯ a, ¯ b, ¯ c$ form a base, find the sum of co-ordinates of the vectors.
$2 u - 3 v + 4 w$ if $u = 2 a + 3 b + 4 c, v = 3 a - 2 b + 4 c, w = 2 a - 4 b + 3 c$

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Solution

Given, $u = 2 a + 3 b + 4 c, v = 3 a - 2 b + 4 c, w = 2 a - 4 b + 3 c$

$2 u = 4 a + 6 b + 8 c$
$3 v = 9 a - 6 b + 12 c$
$4 w = 8 a - 16 b + 12 c$

$\Rightarrow$ $2 u - 3 v + 4 w$
$= (4 - 9 + 8) a + (6 + 6 - 16) b + (12 - 12 + 8) c$
$= 3 a - 4 b + 8 c$

Hence< sum of coordinates $= 8 + 3 - 4$ $= 7$ .

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