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Values of a parameter for which two vectors make an acute (or obtuse) angle for every value of the other variable

The same question, asked 2 times: 28 Jul 2022*, 8 Apr 2024. Every wording below, oldest first.

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  1. Question 1

    28 Jul 2022

    167
    Let SS be the set of all a∈Ra \in R for which the angle between the vectors u→=a(log⁡eb)i^−6j^+3k^\overrightarrow{u}= a\left( \log_{e}b \right)\widehat{i}- 6\widehat{j}+ 3\widehat{k} and v→=(log⁡eb)i^+2j^+2a(log⁡eb)k^,(b>1)\overrightarrow{v}=\left( \log_{e}b \right)\widehat{i}+ 2\widehat{j}+ 2a\left( \log_{e}b \right)\widehat{k},(b > 1) is acute. Then SS is equal to

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  2. Question 2

    8 Apr 2024

    78
    The set of all α\alpha, for which the vector a→=αti^+6j^−3k^\overrightarrow{a}= \alpha t\widehat{i}+ 6\widehat{j}- 3\widehat{k} and  b→=ti^−2j^−2αtk^\ \overrightarrow{b}= t\widehat{i}- 2\widehat{j}- 2\alpha t\widehat{k} are inclined at an obtuse angle for all t∈Rt \in R is :

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