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

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

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

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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.

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