Question

4 vectors are constructed orthogonally to the four faces of general tetrahedron . Each vector is pointed outwards and has length equal to the area of the face . Show that the sum of the four vectors is 0

Answer 1

Answer:

Step-by-step explanation:

So for a general tetrahedron, we can derive that three of the sides are described by the vectors: a⃗ , b⃗ and c⃗ .

We can then write the area vector – using the outward pointing convention – for the first side as:

A⃗ ab=12a⃗ ×b⃗

Similarly, two of the remaining three area vectors can be written:

A⃗ bc=12b⃗ ×c⃗

A⃗ ca=12c⃗ ×a⃗

The final area can be found using the fact that the vectors that define the sides define a closed figure and thus must sum to zero as suggested in the first hint by Blue:

A⃗ (c−a)(b−a)=12(c⃗ −a⃗ )×(b⃗ −a⃗ )

Summing all of these we find that:

12a⃗ ×b⃗ +12b⃗ ×c⃗ +12c⃗ ×a⃗ +12(c⃗ −a⃗ )×(b⃗ −a⃗ )=12(a⃗ ×b⃗ +b⃗ ×c⃗ +c⃗ ×a⃗ +(c⃗ −a⃗ )×(b⃗ −a⃗ ))=0⃗

by applying the properties of distribution, simplifying the cross products and canceling....

Here rectangular box with diagonal is for showing vector...

Hope it helps you..