Two pushes on a box, two legs of a trip, a boat's velocity plus a river's current: vectors keep needing to be added. Arrows can't simply be added like numbers, because directions matter. Splitting each into components turns the problem back into ordinary arithmetic.
Key idea
To add vectors, add their $x$-components and add their $y$-components: $R_x = A_x + B_x$, $R_y = A_y + B_y$. Then $R = \sqrt{R_x^2 + R_y^2}$, and its direction comes from $\tan^{-1}(R_y/R_x)$, checked against the quadrant.
Why it works
Placing the arrows head to tail, the total east-west movement is the sum of each arrow's east-west part, and likewise north-south; the directions along one axis don't interfere with the other. Rebuilding the resultant from its two perpendicular parts is a right triangle again, so the Pythagorean theorem gives its size and inverse tangent gives its angle.
Worked examples
Worked example: At right angles
A drone flies $9\ \text{m}$ east, then $12\ \text{m}$ north. Find the size and direction of its displacement.
- $R_x = 9\ \text{m}$, $R_y = 12\ \text{m}$.
- $R = \sqrt{9^2 + 12^2} = \sqrt{225} = 15\ \text{m}$.
- $\theta = \tan^{-1}\left(\dfrac{12}{9}\right) \approx 53.1^\circ$ north of east.
Answer: $15\ \text{m}$ at about $53.1^\circ$ north of east
Worked example: Any angles
Add $\vec{A}$: $40\ \text{N}$ at $30^\circ$ and $\vec{B}$: $25\ \text{N}$ at $120^\circ$ (angles from $+x$).
- $A_x = 40\cos 30^\circ \approx 34.6$, $A_y = 40\sin 30^\circ = 20.0$.
- $B_x = 25\cos 120^\circ = -12.5$, $B_y = 25\sin 120^\circ \approx 21.7$.
- $R_x \approx 22.1\ \text{N}$, $R_y \approx 41.7\ \text{N}$, so $R \approx 47.2\ \text{N}$ at $\tan^{-1}(41.7/22.1) \approx 62.0^\circ$.
Answer: about $47.2\ \text{N}$ at $62.0^\circ$
Worked example: Watch the quadrant
A resultant has $R_x = -6$ and $R_y = 8$. What is its direction from $+x$?
- A calculator gives $\tan^{-1}(8/-6) \approx -53.1^\circ$, which points into quadrant IV.
- But $R_x < 0$ and $R_y > 0$ puts the vector in quadrant II. Add $180^\circ$: $126.9^\circ$.
Answer: $126.9^\circ$
Common mistake
Adding magnitudes directly, so $3\ \text{N}$ and $4\ \text{N}$ at right angles "make $7\ \text{N}$". That is only true when the vectors point the same way. Check: the resultant is never larger than the sum of the sizes and never smaller than their difference, and it equals the sum only when they are lined up.