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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

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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.

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 and space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space space plus space 4 space straight j with hat on top space minus space straight k with hat on top straight a space equals space square root of 4 plus 1 plus 1 end root space equals space square root of 6 comma space space space space straight b space equals space square root of 9 plus 16 plus 1 end root space equals space square root of 26 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 space straight j with hat on top end cell cell straight k with hat on top end cell row 2 cell negative 1 end cell 1 row 3 4 cell negative 1 end cell end table close vertical bar space equals space left parenthesis 1 minus 4 right parenthesis space straight i with hat on top space minus space left parenthesis negative 2 minus 3 right parenthesis straight j with hat on top space plus space left parenthesis 8 plus 3 right parenthesis straight k with hat on top space equals space minus 3 straight i with hat on top space plus space 5 straight j with hat on top space plus 11 straight k with hat on top therefore space space space space open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar space equals square root of 9 plus 25 plus 1 end root space equals space square root of 155

Now,

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 straight a space straight b space sin space straight theta comma space space space where space straight theta space is space angle space between space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top

therefore space space space space square root of 155 space equals space square root of 6 cross times square root of 26 cross times space sin space straight theta space space space space space space rightwards double arrow space space space space sin space straight theta space equals space square root of 155 over 156 end root

Unit vector perpendicular to

straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space 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 fraction numerator 1 over denominator square root of 155 end fraction left parenthesis negative 3 space straight i with hat on top plus 5 space straight j with hat on top plus 11 space straight k with hat on top right parenthesis

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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