Question

Let A, B, C, D be any four points in space. Prove that `|bar"AB" xx bar"CD" + bar"BC" xx bar"AD" + bar"CA" + bar"BD"|` = 4 (area of triangle ABC).

Answer 1

Let A, B, C, D have position vectors `bar"a", bar"b", bar"c", bar"d"` respectively.

Consider `bar"AB" xx bar"CD" + bar"BC" xx bar"AD" + bar"CA" xx bar"BD"`

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

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

`= bar"b" xx bar"d" - bar"b" xx bar"c" - bar"a" xx bar"d" + bar"a" xx bar"c" + bar"c" xx bar"d" - bar"c" xx bar"a" - bar"b" xx bar"d" + bar"b" xx bar"a" + bar"a" xx bar"d" - bar"a" xx bar"b" - bar"c" xx bar"d" + bar"c" xx bar"b"`

`= bar"b" xx bar"d" - bar"b" xx bar"c" - bar"a" xx bar"d" - bar"c" xx bar"a" + bar"c" xx bar"d" - bar"c" xx bar"a" - bar"b" xx bar"d" - bar"a" xx bar"b" + bar"a" xx bar"d" - bar"a" xx bar"b" - bar"c" xx bar"d" - bar"b" xx bar"c"     ....[because bar"p" xx bar"q" = - bar"q" xx bar"p"]`

`= - 2(bar"a" xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a")`

∴ `|bar"AB" xx bar"CD" + bar"BC" xx bar"AD" + bar"CA" xx bar"BD"|`

= `|- 2(bar"a" xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a")|`

`= 4[1/2 |bar"a"xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a"|]`

= 4(are of Δ ABC).