A kayakeris paddling 2.50 m/s at an angle of 45° (northeast) and the current is moving 1.25 m/s at an angle of 315° (southeast). What is the kayaker's total velocity?

The kayaker has velocity vector

v = (2.50 m/s) (cos(45º) i + sin(45º) j )

v ≈ (1.77 m/s) (i + j )

and the current has velocity vector

w = (1.25 m/s) (cos(315º) i + sin(315º) j )

w ≈ (0.884 m/s) (i - j )

The kayaker's total velocity is the sum of these:

v + w ≈ (2.65 m/s) i + (0.884 m/s) j

That is, the kayaker has a velocity of about ||v + w|| ≈ 2.80 m/s in a direction θ such that

tan(θ) = (0.884 m/s) / (2.65 m/s)   →   θ ≈ 18.4º

or about 18.4º north of east.