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

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

$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 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 4 1 cell negative 3 end cell row 2 1 cell negative 3 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 1 cell negative 3 end cell row 1 cell negative 3 end cell end table close vertical bar space minus straight j with hat on top space open vertical bar table row 4 cell negative 3 end cell row 2 cell negative 3 end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 4 1 row 2 1 end table close vertical bar space space space space space space space space space space space space space space space space space space equals left parenthesis negative 3 plus 3 right parenthesis space straight i with hat on top space minus space left parenthesis negative 12 plus 6 right parenthesis space straight j with hat on top space plus space left parenthesis 4 minus 2 right parenthesis space straight k with hat on top space equals space 6 space straight j with hat on top space plus space 2 space straight k with hat on top therefore space space space space space space open vertical bar straight a with rightwards arrow on top cross times straight b with rightwards arrow on top close vertical bar space equals space square root of 0 plus 36 plus 4 end root space equals space square root of 40 Unit space vector space perpendicular space to space straight a with rightwards arrow on top space and space space straight b with rightwards arrow on top space equals space fraction numerator straight a with rightwards arrow on top cross times straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top 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 equals space plus-or-minus fraction numerator 1 over denominator square root of 40 end fraction left parenthesis 6 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis space equals space plus-or-minus fraction numerator 1 over denominator square root of 10 end fraction left parenthesis 3 space straight j with hat on top space plus space straight k with hat on top right parenthesis 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$

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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.

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