Add three 3-dimensional vector groups together?

Questions
1

When I add two 2-dimensional vector groups together, I get the desired results of two groups. However, when I add three 3-dimensional vector groups together, it appears that each individual group appears as a group - see attached. I am expecting all the 1st elements of the vectors to form a group, and all the 2nd elements of the vectors to form a group and finally all the 3rd elements of the vectors to from a final group, similar to the 2-dimensional vector group. However, I am not sure of the results that I am getting.

Additionally I am getting an error message

"Thread::tdlen: Objects of unequal length in {{{{E^(Plus[<<2>>]
t)},{-Sqrt[2] E^Times[<<2>>]}}},{E^((-2-Power[<<2>>])
t)}}+{<<1>>}+{{{{E^(Plus[<<2>>] t)},{Sqrt[2]
E^(<<5>><<1>><<1>>])}}},{E^<<1>>}} cannot be combined.

I am sure that it is my misunderstanding so please point me in the correct direction.

Thanks,

Mitch Sandlin

3VectorMatrix.nb (9.68 KB)

2

Rewriting the 3-dimensional example this is what I get:

In[1]:= {1, -Sqrt[2], 1} E^((-2 - Sqrt[2]) t) + {-1, 0, 
   1} E^(-2 t) + {1, Sqrt[2], 1} E^((-2 + Sqrt[2]) t)

Out[1]= {-E^(-2 t) + E^((-2 - Sqrt[2]) t) + 
  E^((-2 + Sqrt[2]) t), -Sqrt[2] E^((-2 - Sqrt[2]) t) + 
  Sqrt[2] E^((-2 + Sqrt[2]) t), 
 E^(-2 t) + E^((-2 - Sqrt[2]) t) + E^((-2 + Sqrt[2]) t)}

Looks a bit nicer in a picture:

Vectors.png

No error message