2D Vector Addition — Practice Problem Solutions

1. $v_x=10\cos60°=5.0\text{ m/s}$; $v_y=10\sin60°\approx8.66\text{ m/s}$.

2. Magnitude $=\sqrt{6^2+8^2}=\sqrt{100}=10\text{ km}$; direction $=\arctan\left(\dfrac{8}{6}\right)\approx53.1°$ north of east.

3. $A_x=12\cos25°\approx10.88\text{ m}$, $A_y=12\sin25°\approx5.07\text{ m}$. $B_x=12\cos25°\approx10.88\text{ m}$, $B_y=-12\sin25°\approx-5.07\text{ m}$. Adding: $R_x\approx21.76\text{ m}$, $R_y=0$. The resultant is $21.76\text{ m}$, directed straight along the horizontal (the vertical components cancel exactly since the two vectors are mirror images above and below the horizontal).

4. Magnitude $=\sqrt{7^2+24^2}=\sqrt{625}=25\text{ N}$; direction $=\arctan\left(\dfrac{24}{7}\right)\approx73.7°$ north of east.

5. With east as $+x$ and north as $+y$: $v_x=-15\cos40°\approx-11.49\text{ m/s}$ (negative, since the wind blows toward the west side), $v_y=15\sin40°\approx9.64\text{ m/s}$.

6. Leg 1 (6 km at 50° north of east): $x=6\cos50°\approx3.86\text{ km}$, $y=6\sin50°\approx4.60\text{ km}$. Leg 2 (4 km at 30° west of north): $x=-4\sin30°=-2.0\text{ km}$, $y=4\cos30°\approx3.46\text{ km}$. Totals: $R_x\approx1.86\text{ km}$, $R_y\approx8.06\text{ km}$. a) Magnitude $=\sqrt{1.86^2+8.06^2}\approx8.27\text{ km}$, direction $\approx\arctan(8.06/1.86)\approx77°$ north of east. b) The y-component (north, $\approx8.06\text{ km}$) contributes far more than the x-component ($\approx1.86\text{ km}$) — the resultant is dominated by the northward direction. c) Walking due west instead for leg 2 gives $x=-4\text{ km}$, $y=0$, so the new totals are $R_x\approx-0.14\text{ km}$, $R_y\approx4.60\text{ km}$ — a noticeably smaller magnitude ($\approx4.60\text{ km}$) than part (a). Reasoning: the original 30°-west-of-north leg was still contributing a good amount of northward distance (+3.46 km); a due-west leg contributes none, so the total northward component (and therefore the resultant) shrinks substantially.

7. The resultant's magnitude equals the sum of the two magnitudes when the vectors point in exactly the same direction (0° between them) — their components add fully with nothing lost. The resultant's magnitude is zero when the vectors point in exactly opposite directions (180° apart) — their components cancel completely.