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

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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

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

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 OA with rightwards arrow on top space cross times space OB 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 space straight k with hat on top end cell row cell space 1 end cell cell space 2 end cell cell space 3 end cell row cell negative 3 end cell cell space minus 2 end cell cell space 1 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 2 3 row cell negative 2 end cell 1 end table close vertical bar minus straight j with hat on top space open vertical bar table row cell space 1 end cell cell space 3 end cell row cell negative 3 end cell cell space 1 end cell end table close vertical bar space plus straight k with hat on top space open vertical bar table row 1 cell space space space space space 2 end cell row cell negative 3 end cell cell space space minus 2 end cell end table close vertical bar space space space space space space space space space space space space space space space space equals left parenthesis 2 plus 6 right parenthesis space straight i with hat on top space minus left parenthesis 1 plus 9 right parenthesis space straight j with hat on top space plus space left parenthesis negative 2 plus 6 right parenthesis space straight k with hat on top space equals 8 space straight i with hat on top space minus space 10 space straight j with hat on top space plus space 4 space straight k with hat on top therefore space space space space open vertical bar OA with rightwards arrow on top space cross times space OB with rightwards arrow on top close vertical bar space equals space square root of 64 plus 100 plus 16 end root space equals space square root of 180 space equals space square root of 36 space cross times 5 end root space equals space 6 square root of 5 Area space of space increment OAB space equals space 1 half open vertical bar OA with rightwards arrow on top space cross times space OB with rightwards arrow on top close vertical bar space equals space 1 half 6 square root of 5 space equals space 3 square root of 5 space sq. space units.

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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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