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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

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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.

Let $straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 6 space straight j with hat on top space plus space straight lambda space straight k with hat on top$

(i)    $straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 2 3 cell negative 1 end cell row 4 6 straight lambda end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 3 cell negative 1 end cell row 6 straight lambda end table close vertical bar space minus space straight j with hat on top open vertical bar table row 2 cell negative 1 end cell row 4 straight lambda end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 3 row 4 6 end table close vertical bar$
   $equals left parenthesis 3 straight lambda plus 6 right parenthesis space straight i with hat on top space minus space left parenthesis 2 straight lambda plus 4 right parenthesis straight j with hat on top plus left parenthesis 12 minus 12 right parenthesis straight k with hat on top space equals space left parenthesis 3 straight lambda plus 6 right parenthesis straight i with hat on top space minus space left parenthesis 2 space straight lambda space plus space 4 right parenthesis straight j with hat on top$

Since $straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top$
$therefore space space space left parenthesis 3 straight lambda plus 6 right parenthesis straight i with hat on top space minus space left parenthesis 2 straight lambda plus 4 right parenthesis straight j with hat on top space equals space 0 with rightwards arrow on top therefore space space space 3 space straight lambda space plus space 6 space equals space 0 comma space space space 2 space straight lambda space plus space 4 space equals space 0 space space space rightwards double arrow space space space straight lambda space equals space minus 2$

(ii) Since $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are perpendiuclar
$therefore space space space straight a with rightwards arrow on top space straight b with rightwards arrow on top space equals space 0 therefore space space space space left parenthesis 2 right parenthesis thin space left parenthesis 4 right parenthesis plus left parenthesis 3 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis negative 1 right parenthesis space straight lambda space equals space 8 therefore space space space space 8 plus 18 minus straight lambda space equals space 0 space space space space space space space space space rightwards double arrow space space space space straight lambda space equals space 26$

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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.