If  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 ?


We have,    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
     therefore space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 0 space space space space rightwards double arrow space space space space straight a with rightwards arrow on top. 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
therefore space space space space there space are space three space possibilities colon
(i) straight a with rightwards arrow on top space equals space 0 degree           left parenthesis ii right parenthesis space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals 0 with rightwards arrow on top                (iii) straight a with rightwards arrow on top is perpendicular to straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top
Again,              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
therefore 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 straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space space space rightwards double arrow space space straight a with rightwards arrow on top space cross times space left parenthesis straight b with rightwards arrow on top minus straight c with rightwards arrow on top right parenthesis space equals space 0 with rightwards arrow on top
therefore space space space again space there space are space three space possibilities colon
left parenthesis straight i right parenthesis space space straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space space space left parenthesis ii right parenthesis space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top comma space space space left parenthesis iii right parenthesis space straight a with rightwards arrow on top space is space parallel space to space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top
Now comma space straight a with rightwards arrow on top space is space given space to space be space non minus zero space vector
therefore space space space space we space have space the space following space possibilities space left colon
left parenthesis 1 right parenthesis space straight b with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top
left parenthesis 2 right parenthesis space straight a with rightwards arrow on top space is space perpendicular space to space straight b with rightwards arrow on top minus straight c with rightwards arrow on top space space and space straight a with rightwards arrow on top space is space parllel space to space straight b with rightwards arrow on top minus straight c with rightwards arrow on top comma space which space is space absurd.
therefore space space space space space space only space possibility space left space is space straight b with rightwards arrow on top minus straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top space space or space space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top.

space space space space

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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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