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Question
Find the altitude of a parallelepiped determined by the vectors `vec"a" = - 2hat"i" + 5hat"j" + 3hat"k", vec"b" = hat"i" + 3hat"j" - 2hat"k"` and `vec"c" = - vec"i" + vec"j" + 4vec"k"` if the base is taken as the parallelogram determined by `vec"b"` and `vec"c"`
Solution
Volume = Base Area × Height
`|[(vec"a", vec"b", vec"c")]| = |vec"b" xx vec"c"| xx "Height"`
`|[(vec"a", vec"b", vec"c")]| = |(-2, 5, 3),(1, 3, -2),(-3, 1, 4)|`
= 2(12 + 2) − 5(4 − 6) + 3(1 + 9)
= − 28 + 10 + 30
= 12
`vec"b" xx vec"c" + |(vec"i", vec"j", vec"k"),(1, 3, -2),(-3, 1, 4)|`
= `vec"i"(14) - vec"j"(- 2) + vec"k"(10)`
`vec"b" xx vec"c" = 14vec"i" + 2vec"j" + 10vec"k"`
`|vec"b" xx vec"c"| = sqrt(196 + 4 + 100_`
= `sqrt(300)`
`(sqrt(300)) xx ("Height")` = 12
Height = `12/sqrt(300)`
= `12/(10sqrt(3))`
= `6/(5sqrt(3)) xx sqrt(3)/sqrt(3)`
= `(6sqrt(3))/(5 xx 3)`
= `(2sqrt(3))/5`
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