2D Vector Addition: Finding Your Way Home As the Crow Flies

Is "blocks walked" the same kind of number as "blocks from home"? Same question as distance vs. displacement from the very first lesson of this course — just about to show up in two dimensions instead of one.

Walk 3 blocks east, then 4 blocks north. How far are you from where you started — and in what direction? Not 7 blocks; that's just how far you walked, not how far you ended up. The straight-line distance back to the start is a different number entirely, and finding it is the whole point of two-dimensional vector addition.

Sketch that walk tip-to-tail and it forms a right triangle: 3 blocks along the bottom, 4 blocks up the side, and the straight-line path home is the hypotenuse. The Pythagorean theorem gives its length: $\sqrt{3^2 + 4^2} = 5$ blocks, at an angle of about 53° north of east. That works cleanly because the two legs happen to be perpendicular — but real-world vectors are rarely that convenient, pointing off at arbitrary angles instead of neat 90° turns.

The general tool for any angle is the component method: break each vector into its horizontal (x) and vertical (y) pieces using trigonometry — $x = \text{magnitude} \times \cos\theta$, $y = \text{magnitude} \times \sin\theta$, measuring the angle from due east. Once every vector involved has been split this way, add all the x-pieces together, add all the y-pieces together (separately), and recombine that total x and total y back into a single resultant vector using the Pythagorean theorem for magnitude and inverse tangent for direction. It's the exact same logic as the 3-4-5 city-block example — just formalized so it works for vectors pointing in any direction, not only ones conveniently at right angles.

This is also the same underlying idea used to describe relative velocity between two moving ships or trains, just extended from one dimension into two. And it's about to become essential: describing an object launched at an angle — a thrown ball, a kicked football, a fired projectile — means treating its velocity as one vector with both a horizontal and a vertical component, tracked separately using exactly this method.

Videos

• Introduction to Vector Components — Flipping Physics

Practice Problems

1. A vector has a magnitude of 10 m/s at 60° above the horizontal. Find its x- and y-components.
2. A hiker walks 6 km east, then 8 km north. Find the magnitude and direction of the total displacement.
3. Vector A is 12 m at 25° above the horizontal. Vector B is 12 m at 25° below the horizontal. Find the magnitude and direction of the resultant A + B.
4. A boat experiences a 7 N force east and a 24 N force north at the same time. Find the magnitude and direction of the net force.
5. A wind velocity is 15 m/s directed 40° north of west. Find its x- and y-components. (Use east as positive x, north as positive y.)
6. A hiker walks 6 km at 50° north of east, then 4 km at 30° west of north. a) Find the total displacement (magnitude and direction). b) Which component total (x or y) contributes more to the resultant's magnitude? Justify using your calculated values. c) If the hiker had instead walked the second leg due west (4 km, straight west) instead of 30° west of north, predict whether the total displacement magnitude would increase or decrease compared to part (a). Explain your reasoning using components, without recalculating the exact new magnitude.
7. Two vectors have equal magnitude. Under what condition is their resultant's magnitude equal to the SUM of the two magnitudes (the largest possible resultant)? Under what condition is the resultant's magnitude ZERO (the smallest possible)? Explain both using components.