PYQ Vault
EbooksTutor

Values of a parameter for which the angle between two vectors is acute or obtuse

This type of question, asked 4 times: 28 Jul 2022, 27 Jan 2024, 8 Apr 2024, 22 Jan 2026 S2. All 4 questions below, oldest first.

Try each one first, then open its answer.

  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

    Tap an option to check your answer.

  2. Question 2

    27 Jan 2024

    81
    The least positive integral value of α\alpha, for which the angle between the vectors αi^−2j^+2k^\alpha\widehat{i}- 2\widehat{j}+ 2\widehat{k} and αi^+2αj^−2k^\alpha\widehat{i}+ 2\alpha\widehat{j}- 2\widehat{k} is acute, is.

  3. Question 3

    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 :

    Tap an option to check your answer.

  4. Question 4

    22 Jan 2026 S2

    71
    Let a vector a→=2i^−j^+λk^,λ>0\overrightarrow{a} = \sqrt{2}\widehat{i} - \widehat{j} + \lambda\widehat{k},\lambda > 0, make an obtuse angle with the vector b→=−λ2i^+42j^+42k^\overrightarrow{b} = - \lambda^{2}\widehat{i} + 4\sqrt{2}\widehat{j} + 4\sqrt{2}\widehat{k} and an angle θ,π6<θ<π2\theta,\frac{\pi}{6} < \theta < \frac{\pi}{2}, with the positive zz-axis. If the set of all possible values of λ\lambda is (α,β)−{γ}(\alpha,\beta) - \{\gamma\}, then α+β+γ\alpha + \beta + \gamma is equal to ____\_\_\_\_ .