Revise Vectors: properties and operations for IB Mathematics: Analysis and Approaches (3.7)

A vector is a quantity with both magnitude (size) and direction. In print it is a bold lowercase letter, $\mathbf{v}$; by hand you underline it, $\underline{v}$, or cap it with an arrow, $\vec{v}$. The most useful description is by components — how far the vector reaches along each axis. You can write those components stacked as a column vector, or strung together using the base vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$, which point one unit along the $x$-, $y$- and $z$-axes respectively. The two forms are interchangeable.

A position vector fixes a point relative to the origin $O$: the position vector of $P$ is $\vec{OP}$, and its components are simply the coordinates of $P$. A displacement vector, by contrast, records a journey between two points and does not depend on the origin. The single most important formula in this topic connects them: the vector from $A$ to $B$ is 'finish minus start'.

The vector between two points: $\vec{AB}=\vec{OB}-\vec{OA} \quad (\text{finish} - \text{start}).$ Its length is the distance $AB=|\vec{AB}|$, and reversing gives $\vec{BA}=-\vec{AB}$.

Addition, subtraction and scalar multiplication all work component by component. Geometrically they have a clean picture too. Addition is head-to-tail: to form $\mathbf{a}+\mathbf{b}$, place the tail of $\mathbf{b}$ at the head of $\mathbf{a}$; the resultant runs from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$ (the triangle rule, equivalently the diagonal of a parallelogram). Subtraction $\mathbf{a}-\mathbf{b}$ is $\mathbf{a}+(-\mathbf{b})$, adding the reversed vector. Scalar multiplication $k\mathbf{a}$ stretches the length by $|k|$ and flips the direction when $k<0$.

The magnitude (or modulus) of a vector is its length — a scalar that is never negative. You find it with Pythagoras extended to three dimensions: square each component, add, then take the (positive) square root. A unit vector has magnitude exactly $1$; it captures a direction with the length divided out, and you get it by dividing a vector by its own magnitude. The base vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are themselves unit vectors.

For $\mathbf{v}=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$: $|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}, \qquad \hat{\mathbf{v}}=\dfrac{\mathbf{v}}{|\mathbf{v}|}=\dfrac{1}{|\mathbf{v}|}\mathbf{v}.$ The unit vector $\hat{\mathbf{v}}$ has magnitude $1$ and the same direction as $\mathbf{v}$.

On Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark ($M$) for the correct approach, or an accuracy mark ($A$) for the right value, where an $A$ mark is dependent on the $M$ mark before it. Follow-through (FT) means that a right method applied to a wrong earlier number still earns the marks that depend on it, and the engine accepts any correct exact form. But that protection only exists if the method is on the page. Study how each of the five marks below is earned by a specific line.