Leave lesson

Pure · Vectors

1 / 13

Vector notation and representation

What a vector is (a size and a direction), the ways we write one (bold letter, arrow, column, i–j components), and how to add, subtract and scale vectors — both geometrically (nose-to-tail) and component by component.

Pure · Vectors

Vector notation and representation

What a vector is (a size and a direction), the ways we write one (bold letter, arrow, column, i–j components), and how to add, subtract and scale vectors — both geometrically (nose-to-tail) and component by component.

Why it works

A scalar is just a size — 55, −3-3, a temperature. A vector carries a size and a direction: a displacement of "55 km north-east", a velocity, a force. We draw one as an arrow (a directed line segment): its length is the magnitude and the way it points is the direction.

There are several ways to write the same vector, and you must read all of them:
  • a bold letter a\mathbf{a} (handwritten: underlined, a‾\underline{a});
  • an arrow between two points, AB⃗\vec{AB} — "from AA to BB";
  • a column, (xy)\begin{pmatrix} x \\ y \end{pmatrix};
  • in terms of the unit vectors i\mathbf{i} (one unit along the xx-axis) and
j\mathbf{j} (one unit along the yy-axis): xi+yjx\mathbf{i} + y\mathbf{j}.

The last two say exactly the same thing: (3−2)\begin{pmatrix} 3 \\ -2 \end{pmatrix} is 3i−2j3\mathbf{i} - 2\mathbf{j}. (In three dimensions you add a third unit vector k\mathbf{k} along the zz-axis, giving xi+yj+zkx\mathbf{i} + y\mathbf{j} + z\mathbf{k}.)

Position doesn't matter. Two arrows are the same vector if they have the same length and the same direction — it doesn't matter where on the page they sit. So AB⃗=DC⃗\vec{AB} = \vec{DC} whenever ABAB and DCDC are parallel, equally long and point the same way.

Reversing the arrow flips the sign: BA⃗=−AB⃗\vec{BA} = -\vec{AB} — same length, opposite direction. More generally −a-\mathbf{a} is a\mathbf{a} turned to point backwards.

Scaling. kak\mathbf{a} is a\mathbf{a} stretched by the factor kk: its magnitude is ∣k∣ ∣a∣|k|\,|\mathbf{a}|, it points the same way if k>0k > 0 and the opposite way if k<0k < 0. This is the key fact behind parallel vectors:Two non-zero vectors are parallel exactly when one is a scalar multiple of the other, a=kb\mathbf{a} = k\mathbf{b}.Adding is nose-to-tail (the triangle law): walk along a\mathbf{a}, then along b\mathbf{b} from where you finished, and the single arrow from start to end is a+b\mathbf{a} + \mathbf{b}. In a triangle this reads AB⃗+BC⃗=AC⃗\vec{AB} + \vec{BC} = \vec{AC}: the BB's "cancel".aba + bSubtracting is adding the reverse: a−b=a+(−b)\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b}).

In components it is wonderfully simple — you just work on each direction separately: (a1a2)+(b1b2)=(a1+b1a2+b2),k(a1a2)=(ka1ka2).\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}, \qquad k\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} = \begin{pmatrix} k a_1 \\ k a_2 \end{pmatrix}. Every component gets multiplied by kk — not just the first.