The two adjacent sides of a parallelogram are and Find the u

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Vector Algebra

Short Answer Type

261. Find the area of the parallelogram whose adjacent sides are determined by the vectors

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

and

straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space straight k with hat on top.

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262. Find the area of the parallelogram determined by the vectors

3 space straight i with hat on top space plus space 4 space straight j with hat on top space and space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top.

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263. Find the area of a parallelogram whose adjacent sides are given by the vectors

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

and

straight b with rightwards arrow on top space equals 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

.

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264. Determine the area of the parallelogram, whose adjacent sides are given by the vectors

straight i with hat on top minus straight j with hat on top minus straight k with hat on top space and space space 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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265. Determine the area of the parallelogram, whose adjacent sides are given by the vectors

2 straight i with hat on top plus straight j with hat on top plus straight k with hat on top space and space straight i with hat on top space minus 2 straight j with hat on top plus space straight k with hat on top.

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Long Answer Type

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266. The two adjacent sides of a parallelogram are $2 straight i with hat on top minus 4 straight j with hat on top plus 5 straight k with hat on top$ and $straight i with hat on top space minus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top.$
Find the unit vector parallel to its diagonal. Also find its area.

Let ABCD be the parallelogram with

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

Now

AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space AD with rightwards arrow on top

equals left parenthesis 2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis plus left parenthesis straight i with hat on top minus 2 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis space equals space 3 space straight i with hat on top space minus space 6 space straight j with hat on top space plus space 2 space straight k with hat on top

Here a unit vector parallel to

AC with rightwards arrow on top space equals space fraction numerator 1 over denominator open vertical bar AC with rightwards arrow on top close vertical bar end fraction left parenthesis stack AC right parenthesis with rightwards arrow on top space space space space equals space fraction numerator 1 over denominator square root of 3 squared plus left parenthesis negative 6 right parenthesis squared plus 2 squared end root end fraction left parenthesis 3 straight i with hat on top minus 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space space equals space fraction numerator 1 over denominator square root of 9 plus 36 plus 4 end root end fraction left parenthesis 3 space straight i with hat on top space minus space 6 space straight j with hat on top space space plus space 2 space straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 49 end fraction left parenthesis 3 straight i with hat on top space minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals space 1 over 7 left parenthesis 3 straight i with hat on top space minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals space 3 over 7 straight i with hat on top space minus space 6 over 7 straight j with hat on top space plus space 2 over 7 straight k with hat on top

Vector area of parallelogram ABCD =

AB with rightwards arrow on top space cross times space AD with rightwards arrow on top

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 space space straight k with hat on top end cell row 2 cell negative 4 end cell cell space space 5 end cell row 1 cell negative 2 end cell cell negative 3 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 4 end cell cell space space space space 5 end cell row cell negative 2 end cell cell space minus 3 end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 2 5 row 1 cell negative 3 end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 cell negative 4 end cell row 1 cell negative 2 end cell end table close vertical bar equals space straight i with hat on top left parenthesis 12 plus 10 right parenthesis space minus space straight j with hat on top left parenthesis 6 minus 5 right parenthesis space plus space straight k with hat on top left parenthesis negative 4 plus 4 right parenthesis equals space 22 straight i with hat on top space plus 11 space straight j with hat on top space plus space 0 space straight k with hat on top therefore space area space of space parallelogram space equals space open vertical bar AB with rightwards arrow on top space cross times AD with rightwards arrow on top close vertical bar space equals space open vertical bar 22 space straight i with hat on top space plus space 11 space straight j with hat on top space plus space 0 space straight k with hat on top close vertical bar 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 space space space space equals square root of left parenthesis 22 right parenthesis squared plus left parenthesis 11 right parenthesis squared plus 0 squared end root space equals space square root of 484 plus 121 plus 0 end root 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 space space space equals space square root of 605 space equals space square root of 21 space cross times 5 end root space space equals space 11 square root of 5 space space sq. space units.

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267. Find the area of the parallelogram whose diagonals are

straight a with rightwards arrow on top space equals space 3 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 space space space and space straight b with rightwards arrow on top space equals 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.

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Short Answer Type

268. Find the area of the triangle whose adjacent sides are made by the vectors

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269. Find the area of the triangle formed by O, A, B. when

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

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Long Answer Type

270. Find the vectors area and also the scalar area of a triangle, the position vectors of whose vertices are

negative straight i with hat on top plus straight j with hat on top plus 3 space straight k with hat on top comma space straight i with hat on top plus 2 space straight j with hat on top space plus space 4 space straight k with hat on top space and 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.

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