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Unknown component that makes two vectors perpendicular

This type of question, asked 3 times: 28 Jul 2022, 24 Jan 2023, 23 Jan 2025. All 4 questions below, oldest first.

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  1. Question 1

    28 Jul 2022

    21
    If the projection of 2i^+4j^−2k^2\widehat{i}+ 4\widehat{j}- 2\widehat{k} on i^+2j^+αk^\widehat{i}+ 2\widehat{j}+ \alpha\widehat{k} is zero. Then, the value of α\alpha will be ____\_\_\_\_.

  2. Question 2

    24 Jan 2023

    99
    If two vectors P→=ı^+2mȷ^+mk^\overrightarrow{P}=\widehat{\imath}+ 2m\widehat{\jmath}+ m\widehat{k} and Q→=4ı^−2ȷ^+mk^\overrightarrow{Q}= 4\widehat{\imath}- 2\widehat{\jmath}+ m\widehat{k} are perpendicular to each other. Then, the value of mm will be:

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  3. Question 3

    24 Jan 2023

    21
    Vectors aı^+bȷ^+k^a\widehat{\imath}+ b\widehat{\jmath}+\widehat{k} and 2ı^−3ȷ^+4k^2\widehat{\imath}- 3\widehat{\jmath}+ 4\widehat{k} are perpendicular to each other when 3a+2b=73a + 2b = 7, the ratio of aa to bb is x2\frac{x}{2}. The value of xx is ____\_\_\_\_.

  4. Question 4

    23 Jan 2025

    22
    Two particles are located at equal distance from origin. The position vectors of those are represented by A→=2i^+3nj^+2k^\overrightarrow{A}= 2\widehat{i}+ 3n\widehat{j}+ 2\widehat{k} and B→=2i^−2j^+4pk^\overrightarrow{B}= 2\widehat{i}- 2\widehat{j}+ 4p\widehat{k}, respectively. If both the vectors are at right angle to each other, the value of n−1n^{- 1} is ____\_\_\_\_