A canoe has a velocity of 0.460 m/s southeast relative to the earth. The canoe is on a river that is flowing at 0.600 m/s east relative to the earth. Find the magnitude of the velocity of the canoe relative to the river.

Set up a vector diagram to calculate the velocity of the canoe relative to the river. 
To do this use the relative motion formula.
V_(ac)=V_(ab)+V_(bc)
Put into words, the velocity of a with respect to c is equal to the velocity of a with respect to b plus the velocity of b with respect to b.
You will find:
V_(canoe,river)+V_(river,earth)=V_(canoe,earth)
Where we dont know V_(canoe,river)
We have 2 lengths of the triangle, and the angle between those 2 lengths. For simplicity, lets now call the lengths we do know b and c and the angle between them A. 
The length opposite the angle A is what we require as this is the magnitude of the velocity of the canoe (namely, its speed) relative to the river.
We can calculate the length a using the Cosine Rule:
a^2 = b^2 + c^2 - (2bc cosA)
Giving
a = sqrt (b^2 + c^2 - 2bc cos A)
Now, we have that the lengths b and c are 
b = 0.460 m/s
c = 0.600 m/s
and that the angle A = 45 degrees = pi/4 radians
Finally then we have
a = sqrt ( 0.46^2 + 0.6^2 - (0.46)(0.6)cos (pi/4))
= sqrt ( 0.5716 - 0.276/sqrt(2))
= sqrt (0.3764) = 0.6135 m/s
The magnitude of the canoe relative to the river is 0.614 m/s. However this is westwards on the East/West axis, so that the canoe is drifting southwards and in the opposite direction to the river on the East/West axis relative to the river. You can tell this from your vector diagram. The precise angle of relative drift can be calculated with the Sine Rule or Cosine Rule.
http://hyperphysics.phy-astr.gsu.edu/hbase/relmot.html

https://www.bbc.co.uk/bitesize/guides/zsgjxfr/revision/8