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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.

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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 $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ .

Here $straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top$

$therefore space space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis 6 right parenthesis squared end root space equals square root of 9 plus 4 plus 36 end root space equals square root of 49 space equals 7 therefore space space space open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar space equals space 7 therefore space space space space open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar space space sin space straight theta space equals space 7 comma space space space where space straight theta space is space an space angle space between space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top therefore space space space space space left parenthesis 2 right parenthesis thin space left parenthesis 7 right parenthesis space sin space straight theta space equals space 7 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 2 comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 7 close square brackets therefore space space space sin space straight theta space equals space 1 half space space space rightwards double arrow space space space straight theta space equals space 30 degree$

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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