Question

If four points `"A"(bar"a"), "B"(bar"b"), "C"(bar"c") and "D"(bar"d")` are coplanar, then show that `[(bar"a", bar"b", bar"c")] + [(bar"b", bar"c", bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")]`.

Answer 1

If points `"A"(bar"a"), "B"(bar"b"), "C"(bar"c")` and `"D"(bar"d")` are coplanar, then `bar"AB", bar"AC", bar"AD"` are also coplanar.

∴ `bar"AB"*(bar"AC" xx bar"AD")` = 0        .......(i)

Here, `bar"AB" = bar"b" - bar"a"`

`bar"AC" = bar"c" - bar"a"`

`bar"AD" = bar"d" - bar"a"`

From (i), we get

`(bar"b" - bar"a").[(bar"c" - bar"a") xx (bar"d" - bar"a")]` = 0

∴ `(bar"b" - bar"a").[bar"c" xx bar"d" - bar"c" xx bar"a" - bar"a" xx bar"d" + bar"a" xx bar"a"]` = 0 

∴ `(bar"b" - bar"a")*[bar"c" xx bar"d" - bar"c" xx bar"a" - bar"a" xx bar"d" + bar"a" xx bar0]` = 0

∴ `(bar"b" - bar"a")*[bar"c" xx bar"d" - bar"c" xx bar"a" - bar"a" xx bar"d"]` = 0

∴ `bar"b"*(bar"c" xx bar"d") - bar"b"*(bar"c" xx bar"a") - bar"b"*(bar"a" xx bar"d") - bar"a"*(bar"c" xx bar"d") + bar"a"*(bar"c" xx bar"a") + bar"a"*(bar"a" xx bar"d")` = 0

∴ `[(bar"b", bar"c", bar"d")] - [(bar"b", bar"c", bar"a")] - [(bar"b", bar"a", bar"d")] - [(bar"a", bar"c", bar"d")] + [(bar"a", bar"c", bar"a")] - [(bar"a", bar"a", bar"d")]` = 0

∴ `[(bar"b", bar"c", bar"d")] - [(bar"a", bar"b", bar"c")] + [(bar"a", bar"b", bar"d")] + [(bar"c", bar"a", bar"d")]` + 0 + 0 = 0 

∴ `[(bar"a", bar"b", bar"d")] + [(bar"b", bar"c",bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")]`