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Vector rotated through a right angle in the plane of the vector and an axis

This type of question, asked 2 times: 25 Jun 2022, 25 Jan 2023. All 2 questions below, oldest first.

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  1. Question 1

    25 Jun 2022

    74
    Let a→=a1i^+a2j^+a3k^ ai>0,i=1,2,3\overrightarrow{a}=a_{1}\widehat{i}+a_{2}\widehat{j}+a_{3}\widehat{k}\ a_{i}> 0,i = 1,2,3 be a vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of a→\overrightarrow{a} on the vector 3i^+4j^3\widehat{i}+ 4\widehat{j} be 7 . Let b→\overrightarrow{b} be a vector obtained by rotating a→\overrightarrow{a} with 90∘90^{\circ}. If a→,b→\overrightarrow{a},\overrightarrow{b} and xx-axis are coplanar, then projection of a vector b→\overrightarrow{b} on 3i^+4j^3\widehat{i}+ 4\widehat{j} is equal to

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  2. Question 2

    25 Jan 2023

    64
    The vector a→=−ı^+2ȷ^+k^\overrightarrow{a}= -\widehat{\imath}+ 2\widehat{\jmath}+\widehat{k} is rotated through a right angle, passing through the y-axis in its way and the resulting vector is b→\overrightarrow{b}. Then the projection of 3a→+2b→3\overrightarrow{a}+\sqrt{2}\overrightarrow{b} on c→=5ı^+4ȷ^+3k^\overrightarrow{c}= 5\widehat{\imath}+ 4\widehat{\jmath}+ 3\widehat{k} is:

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