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Find the values of c so that for all real x, the vectors xcijkxci^-6j^+3k^ and xijcxkxi^+2j^+2cxk^ make an obtuse angle. - Mathematics and Statistics

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Sum

Find the values of c so that for all real x, the vectors `"xc"hat"i" - 6hat"j" + 3hat"k"` and `"x"hat"i" + 2hat"j" + 2"cx"hat"k"` make an obtuse angle.

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Solution

Let `bar"a" = "xc"hat"i" - 6hat"j" + 3hat"k"` and `bar"b" = "x"hat"i" + 2hat"j" + 2"cx"hat"k"`

Consider `bar"a".bar"b" = ("xc"hat"i" - 6hat"j" + 3hat"k").("x"hat"i" + 2hat"j" + 2"cx"hat"k")`

`= ("xc")("x") + (-6)(2) + (3)(2"cx")`

`= "cx"^2 - 12 + 6"cx"`

`= "cx"^2 + 6"cx" - 12`

If the angle between `bar"a"` and `bar"b"`  is obtuse, `bar"a".bar"b" < 0`

∴ cx2 + 6cx - 12 < 0

∴ cx2 + 6cx < 12

∴ c(x2 + 6x) < 12

∴ c < `12/("x"^2 + 6"x")`

∴ c < `12/(("x"^2 + "6x" + 9) - 9) = 12/(("x + 3")^2 - 9)`

∴ c < min `{12/(("x + 3")^2 - 9)}` 

Now, `12/(("x + 3")^2 - 9)` is minimum if (x + 3)2 - 9 is maximum

i.e. (x + 3)2 - 9 = ∞ - 9 = ∞

∴ c < min `{12/∞} = 0`

∴ c < 0.

Hence, the angle between `bar"a"` and `bar"b"` is obtuse if c < 0. 

Concept: Vector Product of Vectors (Cross)
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