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Find unknown components of a vector from a given cross product, then a projection

This type of question, asked 2 times: 29 Jun 2022, 27 Jul 2022. All 2 questions below, oldest first.

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  1. Question 1

    29 Jun 2022

    65
    Let  a→=αi^+3j^−k^,b→=3i^−βj^+4k^\ \overrightarrow{a}= \alpha\widehat{i}+ 3\widehat{j}-\widehat{k},\overrightarrow{b}= 3\widehat{i}- \beta\widehat{j}+ 4\widehat{k} and c→=i^+2j^−2k^\overrightarrow{c}=\widehat{i}+ 2\widehat{j}- 2\widehat{k} where α,β∈R\alpha,\beta \in R, be three vectors. If the projection of a→\overrightarrow{a} on c→\overrightarrow{c} is 103\frac{10}{3} and b→×c→=−6i^+10j^+7k^\overrightarrow{b}\times\overrightarrow{c}= - 6\widehat{i}+ 10\widehat{j}+ 7\widehat{k}, then the value of α+β\alpha + \beta equal to:

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  2. Question 2

    27 Jul 2022

    72
    Let a→=αi^+j^+βk^\overrightarrow{a}= \alpha\widehat{i}+\widehat{j}+ \beta\widehat{k} and b→=3i^−5j^+4k^\overrightarrow{b}= 3\widehat{i}- 5\widehat{j}+ 4\widehat{k} be two vectors, such that a→×b→=−i^+9j^+12k^\overrightarrow{a}\times\overrightarrow{b}= -\widehat{i}+ 9\widehat{j}+ 12\widehat{k}. Then the projection of b→−2a→\overrightarrow{b}- 2\overrightarrow{a} on b→+a→\overrightarrow{b}+\overrightarrow{a} is equal to

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