Question

Prove that the vectors u=3i+j-2k ,
v= -i+3j=4k, and w=4i-2j-6k can form the sides of a triangle

They will form a triangle if they are linearly dependent, that is, if
(3,1,-2)= m(-1,3,4) + n(4,-2,-6)

from which we get 3 equations in two unknowns.
-m + 4n = 3 #1
3m - 2n = 1 #2
4m - 6n = -2 #3

let's solve #1 and #2

double #2 plus #1:

-m + 4n = 3
6m - 4n = 2
------------
5m = 5
m=1
back in #1, n=1

substitute those values in #3 which we have not used.
Left side = 4m - 6n
=4 - 6
= -2
= right side

Therefore they are linearly dependent and thus can form a triangle