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Vector Algebra

Short Answer Type

241. Find the unit vector perpendicular to the vectors

straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top space and space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space stack straight j space with hat on top space minus space 3 space straight k with hat on top.

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242. Find the unit vector perpendicular to vectors

3 straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top comma space space space 2 space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top

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243. Find the unit vector perpendicular to vectors

4 space straight i with hat on top space plus space straight j with hat on top space space minus space 3 space straight k with hat on top comma space space space space 2 space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top.

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244. Find a unit vector perpendicular to vectors

3 straight i with hat on top space plus space 2 straight j with hat on top space minus space straight k with hat on top comma space space 2 straight i with hat on top space plus straight j with hat on top space plus 3 straight k with hat on top

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245. Find a unit vector perpendicular to the plane containing the vectors:

straight a with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space minus space 2 straight k with hat on top space space and space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 2 straight j with hat on top space plus space straight k with hat on top

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246. Find a unit vector perpendicular to each of the vector $open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses$ and $left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis$ , where $straight a with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top comma space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top$ .

straight a with rightwards arrow on top space equals straight i with hat on top plus straight j with hat on top plus straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space straight i with hat on top space plus 2 straight j with hat on top space plus space 3 straight k with hat on top

therefore space space space space space space space space space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top comma space space space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space minus straight j with hat on top space minus space 2 space straight k with hat on top therefore space space space space space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space equals space straight i with hat on top open vertical bar table row 3 4 row cell negative 1 end cell cell negative 2 end cell end table close vertical bar minus space straight j with hat on top space open vertical bar table row 2 4 row 0 cell negative 2 end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 3 row 0 cell negative 1 end cell end table close vertical bar 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 equals space left parenthesis negative 6 plus 4 right parenthesis space straight i with hat on top space minus space left parenthesis negative 4 minus 0 right parenthesis space straight j with hat on top space plus space left parenthesis negative 2 minus 0 right parenthesis space straight k with hat on top space equals space minus 2 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 space straight k with hat on top

which is a vector perpendicular to both

open vertical bar left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis close vertical bar space equals space square root of 4 plus 16 plus 4 end root space equals space square root of 24 space equals space square root of 4 space cross times 6 end root space equals space 2 square root of 6

therefore space space space required space unit space vector space space equals space fraction numerator left parenthesis straight a with rightwards arrow on top plus space straight b with rightwards arrow on top right parenthesis thin space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis over denominator open vertical bar left parenthesis straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top right parenthesis space cross times space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis close vertical bar end fraction 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 equals space fraction numerator 1 over denominator 2 square root of 6 end fraction left parenthesis negative 2 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 6 end fraction left parenthesis negative straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top right parenthesis

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Long Answer Type

247.

Find a unit vector perpendicular to each of the vector $straight a with rightwards arrow on top plus straight b with rightwards arrow on top space and space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top comma$ where $straight a with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 2 straight j with hat on top space plus space 2 straight k with hat on top$ and $straight b with rightwards arrow on top space equals space straight i with hat on top space plus space 2 straight j with hat on top space minus space 2 space straight k with hat on top.$

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Short Answer Type

248. Find the value of λ so that the two vectors

2 straight i with hat on top space plus space 3 straight j with hat on top space minus straight k with hat on top

and

4 straight i with hat on top plus 6 straight j with hat on top plus straight lambda straight k with hat on top

are (i) parallel (ii) perpendicular to each other.

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Long Answer Type

249.

Find the value of $straight lambda$ so that the two vectors $2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space straight k with hat on top space space and space 4 space straight i with hat on top space minus space 8 space straight j with hat on top space plus space straight lambda space straight k with hat on top$ are (i) parallel (ii) perpendicular to each other.

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250.

Find the value of $straight lambda$ so that the vectors $2 space straight i with hat on top space plus 4 space straight j with hat on top space minus space 2 space straight k with hat on top$ and $3 straight i with hat on top space plus space 6 straight j with hat on top space plus space straight lambda straight k with hat on top space$ are (i) parallel (ii) perpendicular to each other.

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Short Answer Type

246. Find a unit vector perpendicular to each of the vector and , where .

Long Answer Type

Short Answer Type

Long Answer Type