Answer: R = 37.7 $\hat {i}$ + 16.9 $\hat {j}$

We need to determine the components of the hiker’s displacement for each day.

We can denote the displacement on day 1 as A and day 2 as B.

On Day 1, Displacement A has a magnitude of 25 km and is directed 45$^{\circ}$ below the positive x axis.

$\Rightarrow A_x = A \cos (-\theta) = 25 \times \cos (-45^{\circ}) = 25 \times 0.707 = 17.7\; km$

$\Rightarrow A_y = A \sin (-\theta) = 25 \times \sin (-45^{\circ}) = 25 \times - 0.707 = -17.7\; km$

On Day 2, Displacement B has a magnitude of 40 km and is directed 60$^{\circ}$ above the positive x axis.

$\Rightarrow B_x = B \cos (\theta) = 40 \times \cos (60^{\circ}) = 40 \times 0.5 = 20\; km$

$\Rightarrow B_y = B \sin (\theta) = 40 \times \sin (60^{\circ}) = 40 \times 0.866 = 34.6\; km$

Now, $R_x = A_x + B_x = 17.7 + 20 = 37.7$ and $R_y = A_y + B_y = -17.7 + 34.6 = 16.9$

$\Rightarrow R = 37.7 \hat {i} + 16.9 \hat {j}$