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प्रश्न
The vectors from origin to the points A and B are `vec"a" = 2hat"i" - 3hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + 3hat"j" + hat"k"`, respectively, then the area of triangle OAB is ______.
विकल्प
340
`sqrt(25)`
`sqrt(229)`
`1/2sqrt(229)`
उत्तर
The vectors from origin to the points A and B are `vec"a" = 2hat"i" - 3hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + 3hat"j" + hat"k"`, respectively, then the area of triangle OAB is `1/2sqrt(229)`.
Explanation:
Let O be the origin
∴ `vec"OA" = 2hat"i" - 3hat"j" + 2hat"k"`
And `vec"OB" = 2hat"i" + 3hat"j" + hat"k"`
∴ Area of ΔOAB = `1/2|vec"OA" xx vec"OB"|`
= `1/2|(hat"i", hat"j", hat"k"),(2, -3, 2),(2, 3, 1)|`
= `1/2|hat"i"(-3 - 6) -hat"i"(2 - 4) + hat"k"(6 + 6)|`
= `1/2|-9hat"i" + 2hat"j" + 12hat"k"|`
= `1/2sqrt((-9)^2 + (2)^2 + (12)^2`
= `1/2 sqrt(81 + 4 + 144)`
= `1/2 sqrt(229)`
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