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Question
If `vec"a" = hat"i" - hat"k", vec"b" = xhat"i" + hat"j" + (1 - x)hat"k", vec"c" = yhat"i" + xhat"j" + (1 + x - y)hat"k"`, show that `[(vec"a", vec"b", vec"c")]` depends on neither x nor y
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Solution
Given `vec"a" = vec"i" - vec"k"`
`vec"b" = xvec"i" + vec"j" + (1 - x)vec"k"`
`vec"c" = yvec"i" + xvec"j" + (1 + x - y)vec"k"`
`[(vec"a", vec"b", vec"c")] = |(1, 0, -1),(x, 1, 1 - x),(y, x,11 + x - y)|`
= 1(1 + x – y) – x(1 – x) – 1(x2 - y)
= 1 + x – y – x + x2 – x2 + y
`[(vec"a", vec"b", vec"c")]` = 1
∴ Clearly `[(vec"a", vec"b", vec"c")]` depends on neither x nor y
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