Let A, B, C, D be any four points in space. Prove that ABCDBCADCABD|AB¯×CD¯+BC¯×AD¯+CA¯+BD¯| = 4 (area of triangle ABC). - Mathematics and Statistics

Sum

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).

Solution

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).

Concept: Representation of Vector
Is there an error in this question or solution?