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Determine the value of c so that for all...

Determine the value of `c` so that for all real `x` , vectors `c x hat i-6 hat j-3 hat ka n dx hat i+2 hat j+2c x hat k` make an obtuse angle with each other.

Text Solution

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The correct Answer is:
`c in (-(4)/(3),0)`
Let `vec(a) = xc hat(i) - 6 hat(j) + 3hat(k) " and " vec(b) = xhat(i) + 2hat(j) + 2cxhat(k) " since " vec(a) " and " vec(b)` makes an obtuse angle.
`rArr vec(a)"." vec(b) lt 0 rArr c x^(2) - 12 +6 cx lt0`
`rArr c lt 0 " and " 36c^(2) -4 (-12) c lt 0`
`rArr c gt 0 " and " 12 c (3c + 4) gt 0`
`rArr c gt 0 " and " c gt - 4//3`
` :. c in (-4 //3,0)`
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