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Question
Ler `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = hat"i"` and `vec"c" = "c"_1hat"i" + "c"_2hat"j" + "c"_3hat"k"`. If c1 = 1 and c2 = 2. find c3 such that `vec"a", vec"b"` and `vec"c"` are coplanar
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Solution
Given
`vec"a" = vec"i" + vec"j" + vec"k"`
`vec"b" = vec"i"`
`vec"c" = "c"_1vec"i" + "c"_2vec"j" + "c"_3vec"k"` are coplanar
But c1 = 1 and c2 = 2
So `vec"c" = vec"i" + 2vec"j" + "c"_3vec"k"`
We know that `[vec"a" vec"b" vec"c"]` = 0
`|(1, 1, 1),(1, 0, 0),(1, 2, "c"_3)|` = 0
`1[0] - 1["c"_3] + 1[2]` = 0
⇒ `- "c"_3 + 2` = 0
⇒ c3 = 2
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