Find the area of the triangle whose adjacent sides are made by t

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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

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.

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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

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

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 plus space 4 space straight k with hat on top comma space space space space 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

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 straight j with hat on top end cell cell straight k with hat on top end cell row 3 1 4 row 1 cell negative 1 end cell 1 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 1 4 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 3 4 row 1 1 end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 3 1 row 1 cell negative 1 end cell end table close vertical bar space space space space space space space space space space space space space equals space left parenthesis 1 plus 4 right parenthesis space straight j with hat on top space minus space left parenthesis 3 minus 4 right parenthesis space straight j with hat on top space plus left parenthesis negative 3 minus 1 right parenthesis space straight k with hat on top space equals space 5 space straight i with hat on top space plus space straight j with hat on top space minus 4 space straight k with hat on top space 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 25 plus 1 plus 16 end root space equals space square root of 42

Required area of triangle =

1 half open vertical bar straight a with rightwards arrow on top cross times straight b with rightwards arrow on top close vertical bar space equals space 1 half square root of 42 space sq. space units.

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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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