Vectors — resolving, adding, and the setups IB keeps reusing

Vectors are rarely examined on their own. They are examined inside every mechanics question, every fields question and a good deal of waves — quietly, as a skill the question assumes you already have. Getting a component right is usually the difference between a problem that takes two lines and one that will not come out at all...

Scalar or vector

SCALARS                                                                VECTORS

distance, speed, mass, time, energy,                  displacement, velocity, acceleration, force, work, ,power, temperature,                      momentum, impulse, gravitational, charge                                                                    magnetic field strength

The pairs that catch people out are distance and displacement, speed and velocity, and mass and weight. If a question mentions an object returning to where it started, it is almost certainly testing that displacement is zero while distance is not.

Resolving into components

Any vector can be split into two perpendicular components. For a vector of magnitude F at an angle θ:

A force F resolved into perpendicular components: F cos θ adjacent to the angle and F sin θ opposite it

Resolving a force into components — the component adjacent to the angle takes cosine, the component opposite to the angle takessine.

component adjacent to θ = R cos θ

component opposite to θ = R sin θ

Do not memorise "horizontal is cos". That is only true when θ happens to be measured from the horizontal, and IB regularly measures it from somewhere else specifically to catch that. Look at where the angle actually sits: the component next to the angle takes cosine, the one across from it takes sine. This one habit removes most of the sign and function errors in the whole topic.

Choose your axes to suit the problem rather than the page. On a slope, tilting the axes to run along and perpendicular to the surface turns an awkward problem into an easy one.

Adding vectors

For two vectors, draw them head to tail (or complete the parallelogram) and the resultant runs from the start of the first to the end of the second.

For any number of vectors, use components — it is faster and far less error-prone:

  1. Resolve every vector into x and y components.
  2. Add all the x components; add all the y components.
  3. Recombine.

R = √ ( Rx² + Ry² )
θ = tan⁻¹ ( Ry / Rx )

Then check the quadrant. A calculator's tan⁻¹ only ever returns an angle between −90° and +90°, so if your resultant points into the second or third quadrant the calculator will quietly give you the wrong one. Sketching the resultant roughly before you compute it catches this every time.

The three setups that keep coming back

1 · The inclined plane. Rotate your axes so that x runs down the slope. The weight mg then resolves into mg sin θ down the slope and mg cos θ into the surface.

Block on an inclined plane: weight resolved into mg sin θ down the slope and mg cos θ into the surface, with normal force N

On a slope, the normal force is mg cos θ, not mg — which is why friction is smaller on an incline.

along the slope: mg sin θ

perpendicular: mg cos θ = N

The normal force is mg cos θ, not mg. This is why friction on a slope is smaller than on the flat, and it is the step most often dropped.

2 · Projectile motion. The horizontal and vertical motions are completely independent. Horizontally the velocity is constant at u cos θ; vertically it is ordinary SUVAT with a = 9.81 m s⁻² downwards. The only quantity shared between them is time, which is what makes time the variable to solve for first.

At the top of the flight the vertical velocity is zero but the acceleration is still 9.81 m s⁻² downwards. Setting a = 0 there is a classic and costly error.

3 · Relative velocity. The velocity of A relative to B is a subtraction:

VAB = VA − VB

River crossings and aircraft in crosswinds are nothing more than this. Draw the triangle before you reach for the calculator; almost every mistake in these questions is a drawing mistake, not an arithmetic one.

Where marks are actually lost

  • Using sine where cosine belongs, because the angle was measured from an unfamiliar axis.
  • Taking the normal force on a slope as mg.
  • Adding vector magnitudes as though they were scalars.
  • Trusting tan⁻¹ without checking the quadrant.
  • Setting acceleration to zero at the top of a projectile's path.
  • Answering with a magnitude when the question asked for a vector, and so leaving out the direction

Notes and practice

Lecture notes — Vectors .pdf
Exercises — Vectors .pdf

Next

Uncertainties and error analysis ➡️                                   

Paper 1B: data analysis and graphs ➡️

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