Question

A student adds two displacement vectors that have the magnitudes of 12.0m and 5.0m, with the range of resultant vectors being from 7m to 17m in length. Explain why this is the range of possible resultant vectors.

Answer 1

The point of this question seems to be to draw your attention to the fact that vectors are incompletely specified if their directions are not stated. So adding two incompletely specified quantities cannot lead to a unique answer. 

You can get the idea by drawing a diagram - 

first draw the 12m vector as a line AB in an arbitrary direction (you could equally well start with the 5m vector). 

then, centred at one end (A) of the first vector draw a sphere of radius equal to the magnitude of the second vector. The sphere represents the 5m vector pointing in all possible directions in 3 dimensions. 

Now the sum of the vectors is any line joining the other end (B) of the 1st vector to the surface of the sphere. Clearly the minimum length (magnitude) of the sum is 12 - 5 = 7m, while the maximum length is 12 + 5 = 17m