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")`.

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Answer 1

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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`