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

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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.

Let A (2, b, 3), B (a. – 5, 1) and C (– 1, 11, 9) be given points.

AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals space open parentheses straight a space straight i with hat on top space minus space 5 space straight j with hat on top space plus space straight k with hat on top close parentheses space minus space left parenthesis 2 space straight i with hat on top space plus space straight b space straight j with hat on top plus space 3 space straight k with hat on top right parenthesis space space space space space space equals space left parenthesis straight a minus 2 right parenthesis space straight i with hat on top space minus space left parenthesis straight b plus 5 right parenthesis space straight j with hat on top space space minus space 2 space straight k with hat on top AC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight A space equals space open parentheses negative straight i with hat on top space plus 11 stack straight j space with hat on top plus 9 space straight k with hat on top close parentheses space minus space left parenthesis 2 space straight i with hat on top space plus space straight b space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space space space space space space space equals negative 3 straight i with hat on top space plus space left parenthesis 11 minus straight b right parenthesis space straight j with hat on top space plus space 6 space straight k with hat on top

Since A, B, C are collinear

therefore space space space open vertical bar AB with rightwards arrow on top space cross times space AC with rightwards arrow on top space close vertical bar space equals space 0 with rightwards arrow on top therefore space space 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 cell straight a minus 2 end cell cell negative straight b minus 5 end cell cell negative 2 end cell row cell negative 3 end cell cell 11 minus straight b end cell 6 end table close vertical bar space equals space 0 with rightwards arrow on top rightwards double arrow space space space space open vertical bar table row cell negative straight b minus 5 end cell cell space minus 2 end cell row cell 11 minus straight b end cell cell space space space 6 end cell end table close vertical bar space straight i with hat on top space minus space open vertical bar table row cell straight a minus 2 end cell cell space minus 2 end cell row cell negative 3 end cell cell space space space 6 end cell end table close vertical bar straight j with hat on top space plus space open vertical bar table row cell straight a minus 2 space space space end cell cell negative straight b minus 5 end cell row cell negative 3 end cell cell space 11 minus straight b end cell end table close vertical bar straight k with hat on top space equals 0 with rightwards arrow on top therefore space space space space space left parenthesis negative 6 straight b minus 30 space plus 22 space minus 2 straight b right parenthesis straight i with hat on top space minus space left parenthesis 6 straight a minus 12 minus 6 right parenthesis space straight j with hat on top space plus space open square brackets left parenthesis straight a minus 2 right parenthesis thin space left parenthesis 1 minus straight b right parenthesis space plus 3 space left parenthesis straight b minus 5 right parenthesis close square brackets straight k with hat on top space equals space 0 therefore space space space space minus 6 straight b minus 30 plus 22 minus 2 straight b space equals space 0 comma space space space space 6 straight a minus 12 minus 6 space equals space 0 therefore space space space space space minus 8 straight b space equals space 8 comma space space space 6 straight a space equals 18 rightwards double arrow space space space space space space straight b space equals space minus 1 comma space space space straight a space equals space 3 space space space space space rightwards double arrow space space space space straight a space equals space 3 comma space space space straight b space equals space minus 1

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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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