Scalars and Vectors

Scalar — a quantity with magnitude only. Mass, time, energy, speed, temperature, density.

Vector — a quantity with both magnitude and direction. Displacement, velocity, acceleration, force, momentum.

The distinction has consequences. Walk 3 km north and 4 km south and you have travelled a distance of 7 km, a scalar, but your displacement is 1 km south, because the directions partly cancel.

Adding vectors — draw them nose to tail; the resultant runs from the start of the first to the end of the last. For two perpendicular vectors, Pythagoras and trigonometry give the answer directly:

R = √(A² + B²)

and the angle from

tanθ = opposite ÷ adjacent

Example: forces of 3.0 N east and 4.0 N north give a resultant of √(3.0² + 4.0²) = 5.0 N, at an angle tan⁻¹(4.0/3.0) = 53° north of east. A magnitude without the direction is only half the answer.

Vectors that are not perpendicular — use a scale drawing, or resolve both into perpendicular components and add the components separately.

Resolving a vector — splitting it into two perpendicular components that together have the same effect. For a vector of magnitude F at angle θ to the horizontal:

horizontal component = F cosθ

vertical component = F sinθ

Which is cosine and which is sine — the component adjacent to the angle takes the cosine; the component opposite takes the sine. Rather than memorising it, sketch the triangle and read it off. If the angle is measured from the vertical instead of the horizontal, the two swap, and a sketch catches that immediately.

Example: a 20 N force at 30° above the horizontal has a horizontal component of 20cos30° = 17.3 N and a vertical component of 20sin30° = 10.0 N. Check the result: the horizontal component should be the larger, because 30° is closer to the horizontal than the vertical.

Why resolving is worth doing — perpendicular components are independent. A projectile's horizontal motion is unaffected by gravity acting vertically, which is exactly why projectile problems are solved by treating the two directions separately. Resolving turns one awkward two-dimensional problem into two straightforward one-dimensional ones.