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Repeated cross products with i: simplify and evaluate

The same question, asked 2 times: 1 Feb 2024*, 6 Apr 2024. Every wording below, oldest first.

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  1. Question 1

    1 Feb 2024

    68
    Let a→=−5i^+j^−3k^,b→=i^+2j^−4k^\overrightarrow{a}= - 5\widehat{i}+\widehat{j}- 3\widehat{k},\overrightarrow{b}=\widehat{i}+ 2\widehat{j}- 4\widehat{k} and c→=(((a→×b→)×i^)×i^)×i^\overrightarrow{c}= (((\overrightarrow{a}\times\overrightarrow{b}) \times\widehat{i}) \times\widehat{i}) \times\widehat{i}. Then c→⋅(−i^+j^+k^)\overrightarrow{c}\cdot ( -\widehat{i}+\widehat{j}+\widehat{k}) is equal to

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  2. Question 2

    6 Apr 2024

    167
    Let a→=2i^+j^−k^,b→=((a→×(i^+j^))×i^)×i^\overrightarrow{a}= 2\widehat{i}+\widehat{j}-\widehat{k},\overrightarrow{b}= ((\overrightarrow{a}\times (\widehat{i}+\widehat{j})) \times\widehat{i}) \times\widehat{i}. Then the square of the projection of a→\overrightarrow{a} on b→\overrightarrow{b} is:

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