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Question
If `vec"a", vec"b", vec"c"` are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of `(vec"a" + vec"b") * (vec"b" xx vec"c") + (vec"b" + vec"c")* (vec"c" xx vec"a") + (vec"c" + vec"a") * (vec"a" xx vec"b")`
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Solution
Let `vec"a", vec"b", vec"c"` be the concurrent edges of parallelepiped
Given volume of parallelepiped = 4 cubic units
`[vec"a", vec"b", vec"c"]` = 4
But, `[vec"a", vec"b", vec"c"] = +- 4`
`(vec"a" + vec"b")*(vec"b" xx vec"c") = vec"a"*(vec"b" xx vec"c") + vec"b"*(vec"b" xx vec"c")`
= `[vec"a", vec"b", vec"c"] + [vec"b", vec"b", vec"c"]`
= `[vec"a", vec"b", vec"c"]` + 0
= `[vec"a", vec"b", vec"c"]` .......(1)
Similarly `(vec"b" + vec"c")*(vec"c" xx vec"a") = vec"b"*(vec"c" xx vec"a")`
= `[vec"b", vec"c", vec"a"]`
= `[vec"a", vec"b", vec"c"]` ........(2)
`(vec"c" + vec"a")*(vec"a" xx vec"b") = vec"c"*(vec"a" xx vec"b")`
= `[vec"c", vec"a", vec"b"]`
= `[vec"a",, vec"b", vec"c"]` ........(3)
So, (1) + (2) + (3) = `2[vec"a", vec"b", vec"c"]`
= 3(± 4)
⇒ ± 12
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