Mass, energy and temperature are fully described by a number and a unit. Force, velocity and displacement are not: they carry a direction, and that direction changes everything about how they add. Master the triangle and the right-angle split, and most of mechanics becomes bookkeeping.
A scalar has magnitude only; a vector has magnitude and direction. Coplanar vectors add nose to tail, the resultant running from the start of the first to the end of the last. Any single vector can be split into two perpendicular components, of size F cosθ along one axis and F sinθ along the other, which together carry exactly the same effect as the original.
Set two vectors and watch their resultant form nose to tail, with its perpendicular components drawn in. Compare the resultant with the plain sum of the magnitudes, and notice it only matches when the two point the same way.
Knowing the category is worth easy marks. Each vector below has a scalar partner that is easy to confuse with it.
| Scalar (magnitude only) | Vector (magnitude and direction) |
|---|---|
| distance | displacement |
| speed | velocity |
| mass | weight |
| energy, work | force |
| time, temperature | acceleration, momentum |
To resolve a vector F at angle θ to a chosen axis, the component along that axis is F cosθ and the component perpendicular to it is F sinθ. To find a resultant of two perpendicular components, recombine with R = √(Fx² + Fy²) and direction tanθ = Fy / Fx.
Four quick checks on scalars, vectors, adding and resolving. Each correct answer earns XP and lights this skill on your star map.
Which of the following is a vector quantity?
Two forces of 3.0 N and 4.0 N act at right angles to each other. The magnitude of their resultant is:
A force of 20 N acts at 30° above the horizontal. Its horizontal component is:
In the simulator, two vectors point in different directions. The magnitude of their resultant is always:
Do not add vectors by adding their magnitudes. Two 5 N forces give 10 N only when they point the same way; at right angles the resultant is about 7.1 N, and head-on they cancel to zero. The second classic slip is the cos/sin mix-up when resolving: the component along the axis the angle is measured from uses cosine, and the perpendicular one uses sine. Always sketch the triangle and check which side is adjacent to the angle.
Unlocks once the four checks above are done. Worth more XP, written to AS Paper 1 and 2 standard.
A box of weight 50 N rests on a frictionless slope inclined at 30° to the horizontal. The component of the weight acting down the slope is:
A car travelling east at 12 m s⁻¹ turns and travels north at 12 m s⁻¹. The magnitude of its change in velocity is:
Three coplanar forces act on a point object and it remains in equilibrium. Which statement must be true?
A force of 12 N acts at 60° to the x-axis. Its components along the x and y axes are, respectively:
This skill is now lit gold on your star map. You have finished the lessons of Topic 1; the Paper 1 set awaits.