If  and  then  from Mathematics Vector Algebra

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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.
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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.


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 space space space rightwards double arrow space space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top
rightwards double arrow space space space straight a with rightwards arrow on top space cross times space left parenthesis straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top right parenthesis space equals space 0 with rightwards arrow on top space space space rightwards double arrow space space space straight b with rightwards arrow on top minus straight c with rightwards arrow on top space is space parallel space to space straight a with rightwards arrow on top
rightwards double arrow space space space space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals space straight lambda space straight a with rightwards arrow on top space space space space space rightwards double arrow space space space 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 comma space where space straight lambda space space is 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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