Applied Maths: Units and Vectors (Strand 1) — Higher Level Notes

1. Why this chapter carries the rest of the course

Units and Vectors looks like the easy chapter. It is the one that quietly decides how the next eleven feel. Every projectile question is a vector resolved into components. Every collision is a momentum vector. If resolving a vector is something you do carefully rather than automatically, every later chapter costs you time you will not have in an exam.

Two separate skills live here: dimensional analysis, which is how you check work you have already done, and vector algebra, which is how you do it in the first place.

2. Dimensional analysis: a free checking tool

Every mechanical quantity is built from three fundamentals — mass [M], length [L] and time [T]. Force comes straight out of Newton's second law:

[F] = [M][L][T]^-2

Work is force times distance, so [W] = [M][L]²[T]^-2. Power is work per unit time, so [P] = [M][L]²[T]^-3. You never memorise these. You rebuild them in ten seconds from a definition you already know.

A quantity is dimensionless only when it is a ratio of two quantities of the same kind. The coefficient of friction is a force over a force, so it has no dimensions at all.

3. Scalars and vectors

• Scalar — fully specified by a magnitude and a unit. Mass, speed, work, energy, time.
• Vector — needs a direction too. Displacement, velocity, force, momentum, acceleration.

Two traps follow. Speed is not velocity — speed is the magnitude of the velocity vector and carries no direction. And two vectors of equal magnitude are not equal unless their directions agree as well. A magnitude is never negative.

4. The i-j plane

Write every vector as xi + yj and geometry becomes arithmetic. Magnitude is Pythagoras:

|F| = √(x² + y²)

So 3i + 4j has magnitude 5. Not 7, which is adding the components, and not 25, which is stopping before the square root.

To travel from A to B the displacement is b - a: destination minus start. Reversing that subtraction turns your answer through 180 degrees.

5. Dot products: the perpendicular test

The dot product multiplies matching components and adds them:

a · b = a₁b₁ + a₂b₂ = |a||b| cosθ

Two consequences do most of the work in exam questions:

• Perpendicular vectors have a dot product of zero. Test 2i + 6j against 9i - 3j: 18 - 18 = 0, so they are perpendicular. Settled in one line, no diagram needed.
• The angle between two vectors comes from rearranging the same identity. You cannot read it off the components of one vector alone.

A vector perpendicular to xi + yj is -yi + xj — swap the components, change one sign. Simply reversing the vector gives something antiparallel, which is a different thing.

6. Polar form

A vector of magnitude r at angle θ anticlockwise from the positive x-axis resolves as:

r cosθ i + r sinθ j

Mind the quadrant. Between 90 and 180 degrees the cosine is negative while the sine is still positive, so the i component is negative and the j component is not. Making both negative because one is negative is a standard slip.

7. Distance is not displacement

Distance accumulates along the path you actually walked. Displacement is the straight line from start to finish, and it has a direction.

Go 300 m east then 400 m north: distance 700 m, displacement 500 m — the hypotenuse of a 3-4-5 triangle. Run three laps of a 400 m track: distance 1200 m, displacement zero, because you finished where you started.

The same split applies to their rates. Average speed is total distance over time; average velocity is displacement over time. They agree only when the motion never turns.