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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

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252. Find a vector of magnitude 19 and which is perpendicular to both the vectors $4 space straight i with hat on top space minus space straight j with hat on top space plus space 8 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus straight j with hat on top space plus space straight k with hat on top$

Let $straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space straight j with hat on top space plus space 8 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus straight j with hat on top space plus space straight k with hat on top$
$straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 4 cell negative 1 end cell 8 row 0 cell negative 1 end cell 1 end table close vertical bar space equals straight i with hat on top space open vertical bar table row cell negative 1 end cell 8 row cell negative 1 end cell 1 end table close vertical bar space minus straight j with hat on top space open vertical bar table row 4 8 row 0 1 end table close vertical bar plus space straight k with hat on top open vertical bar table row 4 cell negative 1 end cell row 0 cell negative 1 end cell end table close vertical bar$
$equals left parenthesis negative 1 plus 8 right parenthesis space straight i with hat on top space minus left parenthesis 4 minus 0 right parenthesis space straight j with hat on top plus left parenthesis negative 4 plus 0 right parenthesis space straight k with hat on top space equals space 7 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top$
which is vector perpendicular to $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ .
$open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 7 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root space equals space square root of 49 plus 16 plus 16 end root space equals square root of 81 space equals space 9$
$therefore$ a unit vector perpendicular to $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ is
$equals space fraction numerator straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar end fraction space equals space 1 over 9 left parenthesis 7 straight i with hat on top space minus 4 space straight j with hat on top space minus space 4 straight k with hat on top right parenthesis$
$therefore$ a vector of magnitude of 19 and perpendicular to $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ is
$equals 19 space open parentheses fraction numerator straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top over denominator open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar end fraction close parentheses space equals 19 over 9 left parenthesis 7 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis$

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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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