Preview Activity 9.4.1.
(a)
Our first task is to understand what it means to complete a right-handed coordinate system. For each of the cases below, you need to give a vector that fills in the blank to create a right-handed coordinate system. For example, the answer that would complete \(\{\vi, \vj, ???\}\) would be \(\vk\text{.}\) You should be careful to when you have vectors that are in the opposite directions of \(\vi\) or \(\vj\) or \(\vk\text{.}\)
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\(\displaystyle \{\vj,\vi, ???\}\)
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\(\displaystyle \{-\vi,\vj, ???\}\)
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\(\displaystyle \{\vk,\vi, ???\}\)
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\(\displaystyle \{-\vi,-\vj, ???\}\)
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\(\displaystyle \{\vk,\vj, ???\}\)
(b)
Explain why there is not a way to complete \(\{\vi,\vi, ???\}\) to be a right-handed coordinate system.
(c)
As we said above, we would like the cross product to distribute over vector sums (\((\vv+\vu) \times \vw = (\vv \times \vw) + (\vu \times \vw)\)). Complete each of the following right-handed coordinate systems geometrically. Focus on the direction of the vector needed to complete the right-handed coordinate system and do not worry about the length of the vector you choose.
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\(\displaystyle \{\vi,\vk, ???\}\)
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\(\displaystyle \{\vj,\vk, ???\}\)
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\(\displaystyle \{\vi+\vj,\vk, ???\}\)
Be sure to note how your thumb is oriented in the last case and verify that the sum of the first two results gives you the last case.
(d)
Finally, we want to address the relationship between the cross product and scalar multiplication.
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What direction should the third vector in the right-handed coordinate system \(\{2\vi,3\vj, ???\}\) be?
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What direction should the third vector in the right-handed coordinate system \(\{2\vk,\vj, ???\}\) be?
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Note that \(3\vk=\vk+\vk+\vk\text{,}\) so \((3\vk) \times \vj = (\vk \times \vj) + (\vk \times \vj) + (\vk \times \vj)\text{.}\) What do you think the vector \((3\vk) \times \vj\) will be?
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What do you think the result of \((4\vi)\times (-6\vj)\) will be?
