01 / CHAPTER 07
Vectors and components
Turn a two- or three-dimensional motion into coupled scalar problems.
ANCHOR EQUATION
Vector position
In symbols: r = xi + yj + zk
Components let one vector motion problem become coupled scalar problems.
Components are independent bookkeeping
A vector equation can be projected onto perpendicular axes without changing the physics. The benefit is that a two-dimensional motion becomes two scalar equations that share the same time variable.
Resolve before solving
For a vector A making angle θ with +x, its components are Aₓ = A cosθ and Aᵧ = A sinθ. Apply this to r, v, and a. If the axes are orthonormal, the vector equation r = x i + y j is equivalent to the pair x and y; no information is lost because x and y can reconstruct r.
A 10 N force at 30° has Fₓ = 10 cos30° = 8.66 N and Fᵧ = 10 sin30° = 5.00 N. Squaring and adding gives √(8.66² + 5.00²) ≈ 10 N, a quick component check.
Bad axes create hard algebra, not new physics
Components are not independent if a constraint or force couples them; they are merely separate equations. Choosing axes along a slope or along symmetry can simplify the equations, but changing axes does not alter measurable acceleration.
Work it: A 2.0 kg block is pulled by 12 N at 40° on a frictionless table. Compute its acceleration components and magnitude; then repeat for a 20° axis rotation to see which answer is invariant.