If       then prove that   where m is a scalar. from Math

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 Multiple Choice QuestionsShort Answer Type

281. If either straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space or space space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space then space space space space straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top. Is the  converse true? Justify your answer with an example.
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282.

If   straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space show that straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top cross times straight c with rightwards arrow on top space equals straight c with rightwards arrow on top cross times straight a with rightwards arrow on top. Interpret the result geometrically.

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283. If straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space not equal to space 0 with rightwards arrow on top comma      then prove that  straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top space equals space straight m space straight b with rightwards arrow on top where m is a scalar.


straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space straight b with rightwards arrow on top cross times straight c with rightwards arrow on top space space space space space rightwards double arrow space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus straight c with rightwards arrow on top space cross times space straight b with rightwards arrow on top space space space space space rightwards double arrow space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top
rightwards double arrow space space space space space left parenthesis straight a with rightwards arrow on top cross times straight c with rightwards arrow on top right parenthesis space cross times space straight b with rightwards arrow on top space equals 0 with rightwards arrow on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis
because space space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space space equals space straight b with rightwards arrow on top space cross times straight c with rightwards arrow on top space not equal to 0 with rightwards arrow on top space
therefore space space space space none space of space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top space space is space straight a space zero space vector
therefore space space space from space left parenthesis 1 right parenthesis comma space it space follows space that space straight a with rightwards arrow on top plus straight c with rightwards arrow on top space is space parallel space to space straight b with rightwards arrow on top
therefore space space space space straight a with rightwards arrow on top plus straight c with rightwards arrow on top space equals space straight m space straight b with rightwards arrow on top space space where space straight m space is space some space scalar. space space space space space space space space space space space space space space space space space space space space space space space space space space
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284.

If straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals straight a with rightwards arrow on top cross times straight c with rightwards arrow on top and  straight a with rightwards arrow on top space not equal to space 0 with rightwards arrow on top comma then straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space plus space straight lambda space straight a with rightwards arrow on top, where straight lambda is a scalar.

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285.

If straight a with rightwards arrow on top not equal to 0 with rightwards arrow on top comma space space do comma space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top. space straight c with rightwards arrow on top space and space space straight a with rightwards arrow on top space cross times straight b with rightwards arrow on top space equals straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space together space imply space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top ?

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286.

If straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top and straight a with rightwards arrow on top not equal to 0 with rightwards arrow on top comma space then straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space plus space straight lambda space straight alpha with rightwards arrow on top comma space where space straight lambda space is space straight a space scalar.

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287.

If straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top  are mutually perpendicular unit vectors and straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top comma then straight b with rightwards arrow on top space equals straight c with rightwards arrow on top space cross times straight a with rightwards arrow on top space and space straight a with rightwards arrow on top space equals space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top.

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288.

If straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space cross times space straight d with rightwards arrow on top space and space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top space cross times space straight d with rightwards arrow on top comma space show space that space straight a with rightwards arrow on top space minus space straight d with rightwards arrow on top is parallel to straight b with rightwards arrow on top minus straight c with rightwards arrow on top space provided space straight a with rightwards arrow on top space not equal to space straight d with rightwards arrow on top space and space straight b with rightwards arrow on top not equal to straight c with rightwards arrow on top.

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289. Show that left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals space 2 space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top .
OR
Show that left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals space 2 space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top and give a geometrical interpretation of it.
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 Multiple Choice QuestionsLong Answer Type

290.

If straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top are three vectors such that straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top comma space space space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space show that the three vectors straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top are orthogonal in pairs and open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top close vertical bar comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals 1.

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