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Question
Determine whether the three vectors `2hat"i" + 3hat"j" + hat"k", hat"i" - 2hat"j" + 2hat"k"` and `3hat"i" + hat"j" + 3hat"k"` are coplanar
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Solution
Let `vec"a" = 2vec"i" + vec"j" + vec"k"`
`vec"b" = vec"i" - 2vec"j" + 2vec"k"`
`vec"c" = 3vec"i" + vec"j" + 3vec"k"`
We know that `vec"a", vec"b", vec"c"` are coplanar if and only if `[vec"a", vec"b", vec"c"]` = 0
`[vec"a", vec"b", vec"c"] = |(2, 3, 1),(1, -2, 2),(3, 1, 3)|`
= `2(- 6 - 2) - 3(3 - 6) + (1 + 6)`
= −16 + 9 + 7
= 0
∴ Gives vectors are coplanar
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