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Question
Find the area of the parallelogram whose two adjacent sides are determined by the vectors `hat"i" + 2hat"j" + 3hat"k"` and `3hat"i" - 2hat"j" + hat"k"`
Solution
Let the given vectors be `vec"a" = hat"i" + 2hat"j" + 3hat"k"`
`vec"b" = 3hat"i" - 2hat"j" + hat"k"`
`vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(1, 2, 3),(3, -2, 1)|`
= `hat"i"(2 + 6) - hat"j"(1 - 9) + hat"k"(-2 - 6)`
= `8hat"i" + 8hat"j" - 8hat"k"`
`|vec"a" xx vec"b"| = sqrt(8^2 + 8^2 + (-8)^2`
= `sqrt(3 xx 8^2)`
= `8sqrt(3)`
The area of the parallelogram with adjacent sides `vec"a"` and `vec"b"`
A = `|vec"a" xx vec"b"|`
= `8sqrt(3)` sq.units
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