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

To prove that the vectors u, v, and w can form the sides of a triangle, we need to show that they are linearly dependent. In other words, we need to find scalars m and n such that u = m*v + n*w.

We can write the equation as (3,1,-2) = m*(-1,3,4) + n*(4,-2,-6).

This gives us three equations:
-m + 4n = 3 (equation 1)
3m - 2n = 1 (equation 2)
4m - 6n = -2 (equation 3)

We can solve equations 1 and 2 to find the values of m and n.

To eliminate the variables m and n, we can multiply equation 2 by 2 and then add it to equation 1:

-m + 4n = 3
6m - 4n = 2
--------------
5m = 5

Dividing both sides by 5, we get m = 1.

Now, substitute m = 1 into equation 1:

-1 + 4n = 3
4n = 4
n = 1

So, we have found the values of m and n as 1.

Now, substitute these values into equation 3 and check if both sides are equal:

Left side = 4m - 6n
= 4 - 6
= -2

Right side = -2

Since the left side is equal to the right side, the vectors u, v, and w are linearly dependent. Hence, they can form the sides of a triangle.