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Vector Algebra

Long Answer Type

251. Find a vector of magnitude 9 and which is perpendicular to both the vectors
$straight a with rightwards arrow on top space equals space 4 straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus 2 space straight i with hat on top space space plus space straight j with hat on top space minus space 2 space straight k with hat on top$

Here $straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space space straight b with rightwards arrow on top space equals space minus 2 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top$
$therefore space straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 4 cell negative 1 end cell 3 row cell negative 2 end cell cell negative 1 end cell cell negative 2 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 1 end cell 3 row 1 cell negative 2 end cell end table close vertical bar space minus straight j with hat on top space open vertical bar table row 4 3 row cell negative 2 end cell cell negative 2 end cell end table close vertical bar plus space straight k with hat on top space open vertical bar table row 4 cell negative 1 end cell row 2 cell negative 1 end cell end table close vertical bar space space space space space space space space space space space space space space space space equals space left parenthesis 2 minus 3 right parenthesis space straight i with hat on top space minus space left parenthesis negative 8 plus 6 right parenthesis space straight j with hat on top space plus space left parenthesis 4 minus 2 right parenthesis space straight k with hat on top space equals space minus straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top$
which is vector perpendicular to $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ .
$open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space square root of 1 plus 4 plus 4 end root space equals space square root of 9 space equals space 3$
$therefore space space space$ a unit vector perpendicular to $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ is $equals space fraction numerator straight a with rightwards arrow on top cross times straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar end fraction space equals 1 third left parenthesis negative straight i with hat on top plus 2 space straight j with hat on top space plus space space 2 space straight k with hat on top right parenthesis$

$therefore$ a vector of magnitude 9 and perp. to $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ is
$equals space 9 space open parentheses fraction numerator straight a with rightwards arrow on top cross times straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar end fraction close parentheses space equals 9 over 3 left parenthesis negative straight i with hat on top space plus space 2 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis space equals space 3 space left parenthesis negative straight i with hat on top space plus space 2 straight j with hat on top space plus 2 straight k with hat on top right parenthesis$
Required vector can be = $negative 3 left parenthesis negative straight i with hat on top space plus 2 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis.$

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252. Find a vector of magnitude 19 and which is perpendicular to both the vectors

4 space straight i with hat on top space minus space straight j with hat on top space plus space 8 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus straight j with hat on top space plus space straight k with hat on top

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253. Find a vector of magnitude 5 units, perpendicular to each of the vectors

left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis space and space left parenthesis straight a with rightwards arrow on top space minus 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 plus straight j with hat on top plus 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 plus 3 straight k with hat on top.

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Short Answer Type

254. Find the sine of the angle between the vectors

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

. Also find a unit vector perpendicular to both the vectors.

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255. Find the sine of the angle between the vectors

straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top space and space straight b with rightwards arrow on top space equals 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 5 space straight k with hat on top

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256.

If $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are two vectors such that $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 2 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 7$ and $straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top space semicolon$ find the angle between $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ .

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257.

If $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 5 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 4 space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 16 comma space find space open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar.$

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258.

Find $open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar$ if $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 10 comma space space space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 2 space space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 12.$

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Long Answer Type

259. Use vector method only to find the values of a and b if points (2, b, 3), (a, – 5, 1) and ( – 1, 11, 9) are collinear.

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Short Answer Type

260. Find the area of the parallelogram having two adjacent sides OA and OB where O is the origin and the position vectors of A and B are respectively

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

and

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

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