Read vector types and notation

Use a displacement for direction and a position vector for location. Normalise a nonzero vector by dividing by its magnitude. Keep signed projections separate from lengths.

Read vector types and notation

Start by asking what the quantity describes: an amount, a location, or a directed change. Distance is a scalar; displacement needs both a length and a direction. Use the vocabulary below to distinguish these roles before choosing a formula.

The source introduces path integration through ants: model each leg of a zig-zag route by a displacement vector and add the legs to find the resultant. Mathematically, the direct return displacement is the negative of that sum. The ant photograph illustrates this motivation; it does not impose numerical data on a vector problem.

A scalar has magnitude only. The printed examples are mass, area, volume, earthquake strength, pollution index, energy, absolute temperature, distance and speed, represented by a number with suitable units. A vector has magnitude and direction: displacement, velocity, acceleration, force, momentum and torque are the source examples. Force magnitude is measured in newtons; adding forces must account for direction as well as magnitude.

The opening outline covers addition/subtraction/scaling and their geometry; position/displacement/direction; modulus, unit vectors and distance; collinearity and the ratio theorem; scalar/vector products, their properties and angles; and the geometric meanings of $\lvert\mathbf a\cdot\widehat{\mathbf n}\rvert$ and $\lvert\mathbf a\times\widehat{\mathbf n}\rvert$. This reading begins with the vocabulary and basic algebra. The source’s QR opens its O-Level vectors refresher.

Geometrical representation

A directed line segment: its length is the magnitude and its arrow gives direction. The source uses bold letters in print and underlined letters in handwriting.

Modulus

$\lvert\mathbf a\rvert$ is the vector’s magnitude or length, a scalar.

Zero/null vector

$\mathbf0$ has magnitude 0 and can be represented by a point. It has no direction.

Parallel vectors

Nonzero $\mathbf a,\mathbf b$ are parallel iff $\mathbf a=k\mathbf b$ for a nonzero real k. A positive k preserves direction; a negative k reverses it.

Negative vector

$-\mathbf a$ has the same magnitude and opposite direction when $\mathbf a\ne\mathbf0$. For zero, negation leaves zero unchanged.

Unit vector

$\widehat{\mathbf a}=\mathbf a/\lvert\mathbf a\rvert$ for nonzero a; its magnitude is 1.

Free vector

Only magnitude and direction are specified; its representation has no fixed initial or terminal point.

Position vector

$\overrightarrow{OA}$ locates A relative to a chosen origin O.

Displacement vector

$\overrightarrow{AB}$ starts at A and ends at B. It can start at any point; a position vector is the special case that starts at O.

Direction vector

The displacement between two distinct points on a line, or any nonzero scalar multiple of it, supplies a direction for that line.

Keep location separate from displacement

The printed displacement row describes a starting point other than O; the general definition also permits O. Translating a free-vector arrow preserves the vector, but changing the origin changes a point’s position vector.

Scale a unit direction to the required length

Normalisation also gives $\mathbf a=\lvert\mathbf a\rvert\widehat{\mathbf a}$. A vector b parallel to a and in the same direction is $\mathbf b=\lvert\mathbf b\rvert\widehat{\mathbf a}$: the source example of magnitude 6 is $6\widehat{\mathbf a}$. A vector in the opposite direction uses a negative multiple. Zero cannot be normalised.