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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.

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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.

Let $straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top minus space 2 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top plus 6 space straight j with hat on top space plus space straight lambda space straight k with hat on top$
$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 4 cell negative 2 end cell row 3 6 cell space space straight lambda end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 4 cell negative 2 end cell row 6 cell space space straight lambda end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 2 cell negative 2 end cell row 3 cell space space space straight lambda end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 4 row 3 6 end table close vertical bar space space space space space space space space space space space space equals space left parenthesis 4 straight lambda plus 12 right parenthesis space straight i with hat on top space minus space left parenthesis 2 straight lambda plus 6 right parenthesis straight j with hat on top space plus space left parenthesis 12 minus 12 right parenthesis straight k with hat on top space equals space left parenthesis 4 straight lambda plus 12 right parenthesis straight i with hat on top space minus space left parenthesis 2 straight lambda plus 6 right parenthesis straight j with hat on top Since space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space are space parallel therefore space space space straight a 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 therefore space space space space left parenthesis 4 space straight lambda space plus 12 right parenthesis space straight i with hat on top space minus space left parenthesis 2 straight lambda plus 6 right parenthesis straight j with hat on top space equals 0 with rightwards arrow on top rightwards double arrow space space 4 straight lambda plus 12 space equals space 0 comma space space 2 straight lambda plus 6 space equals space 0 space space space rightwards double arrow space space space space straight lambda space equals space minus 3$
(ii) Since $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are perpendicular
$therefore space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 therefore space space space left parenthesis 2 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis negative 2 right parenthesis space straight lambda space equals space 0 therefore space space space 6 plus 24 minus 2 straight lambda space equals space 0 space space space space space space space space space space space space space space space space space space space rightwards double arrow space 2 space straight lambda space equals 30 space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight lambda space equals space 15$