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Question
The value of `hat"i"*(hat"j" xx hat"k") + hat"j"*(hat"i" xx hat"k") + hat"k"*(hat"i" xx hat"j")`.
Options
0
−1
1
3
Solution
1
Explanation:
`hat"i"*(hat"j" xx hat"k") + hat"j"*(hat"i" xx hat"k") + hat"k"*(hat"i" xx hat"j")` ...(Given)
Using the standard vector cross product rules:
`hatj xx hatk = hati`
Using the cyclic property of the cross product:
`hati xx hatk = -hatj`
Using the cyclic property:
i^×j^=k^.\hat{i} \times \hat{j} = \hat{k}.
`hati xx hatj = hatk`
∴ `hat"i"*(hat"j" xx hat"k") + hat"j"*(hat"i" xx hat"k") + hat"k"*(hat"i" xx hat"j")`
= `hati * hati + hatj * (-hatj) + hatk * hatk` ...(i)
∴ `hati * hati = 1`
`hatj * (-hatj) = -1`
`hatk * hatk = 1`
Substituting this values in equation (i), we get
1 + (−1) + 1 = 1
∴ `hat"i"*(hat"j" xx hat"k") + hat"j"*(hat"i" xx hat"k") + hat"k"*(hat"i" xx hat"j") = 1`
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