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Question
If `bara = hati - 2hatj`, `barb = hati + 2hatj, barc = 2hati + hatj - 2hatk`, then find (i) `bara xx (barb xx barc)` (ii) `(bara xx barb) xx barc`. Are the results same? Justify.
Solution
(i) `bara xx (barb xx barc)`
`barb xx barc = |(hati, hatj, hatk),(1, 2, 0),(2, 1, -2)|`
= `(-4 - 0)hati - (-2 - 0)hatj + (1 - 4)hatk`
= `-4hati + 2hatj - 3hatk`
∴ `bara xx (barb xx barc) = |(hati, hatj, hatk),(1, -2, 0),(-4, 2, -3)|`
= `(6 - 0)hati - ( -3 - 0)hatj + (2 - 8)hatk`
= `6hati + 3hatj - 6hatk`
(ii) `(bara xx barb) xx barc`
`bara xx barb = |(hati, hatj, hatk),(1, -2, 0),(1, 2, 0)|`
= `(0 - 0)hati - (0 - 0)hatj + (2 - (-2))hatk`
= `4hatk`
∴ `(bara xx barb) xx barc = |(hati, hatj, hatk),(0, 0, 4),(2, 1, -2)|`
= `(0 - 4)hati - (0 - 8)hatj + (0 - 0)hatk`
= `-4hati + 8hatj`
`bara xx (barb xx barc) ≠ (bara xx barb) xx barc`
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