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Question
If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k"` and `"c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, prove that c is the geometric mean of a and b
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Solution
Let `vec"a"_1 = "a"vec"i" + "a"vec"j" + "c"vec"k"`
`vec"a"_2 = vec"i" + vec"k"`
`vec"a"_3 = "c"vec"i" + "c"vec"j" + "b"vec"k"`
But `vec"a"_1, vec"a"_2, vec"a"_3` are coplanar .......(Given)
So, `[(vec"a"_1, vec"a"_2, vec"a"_3)]` = 0
`|("a", "a", "c"),(1, 0, 1),("c", "c", "b")|` = 0
⇒ `"a"[0 - "c"] - "a"["b" - "c"] + "c"["c" - 0]` = 0
⇒ `- "ac" - "Ab" + "ac" + "c"^2` = 0
⇒ c2 = ab
⇒ c = `sqrt("ab")`
∴ c is the geometric means of ‘a’ and ‘b’.
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