Vectors (intro)
The idea
A vector packages magnitude and direction into a single object — a displacement of 6 km northeast is a vector, while the bare 6 km is a scalar. In coordinates, a vector is written by components, like v = (4, 1), recording movement along each axis; addition stacks displacements tip-to-tail by adding components, and multiplying by a scalar stretches or flips the vector. Magnitude comes from the Pythagorean theorem, |v| = √(x² + y²), and direction from trigonometry — vectors are where your coordinate, triangle, and slope skills converge.
Translate every vector problem into components, compute there, and convert back to magnitude-and-direction only at the end; components turn geometry into arithmetic. The central misconception is adding magnitudes: walking 4 km and then 3 km does NOT generally leave you 7 km from the start — only perfectly aligned legs add lengths, and perpendicular legs give √(4² + 3²) = 5 km. Magnitudes obey the triangle inequality, not simple addition. When reporting a direction, anchor it to a stated reference, such as an angle north of east, so the answer cannot be misread.
Worked example
A hiker's morning leg is the displacement (4, 1) km — 4 km east and 1 km north — and the afternoon leg is (2, 5) km. Find the total displacement, its magnitude, and its direction as an angle north of east.
- Add the legs componentwise: total = (4 + 2, 1 + 5) = (6, 6) km, meaning the hiker ends 6 km east and 6 km north of the start.
- Magnitude by the Pythagorean theorem: |total| = √(6² + 6²) = √72 = 6√2 ≈ 8.49 km.
- Direction from the components: tan θ = north/east = 6/6 = 1, so θ = tan⁻¹(1) = 45° north of east.
- Contrast displacement with distance walked: the legs measure √(16 + 1) = √17 ≈ 4.12 km and √(4 + 25) = √29 ≈ 5.39 km, totaling about 9.51 km of walking — more than the 8.49 km displacement, exactly as the triangle inequality demands.
Answer. The total displacement is (6, 6) km: magnitude 6√2 ≈ 8.49 km, directed 45° north of east.
Check your understanding
- Why does adding vectors componentwise agree with the tip-to-tail picture of stacking displacements?
- Under what condition does the magnitude of a sum equal the sum of the magnitudes, and what does the triangle inequality say in every other case?
- How do scalars and vectors differ in the information they carry, and which everyday quantities belong to each kind?
- What does multiplying a vector by −2 do to its magnitude and its direction, and why?
Build the foundations first
Vectors (intro) builds on these concepts. If any feel shaky, start there.