Vectors in the plane

Vector, components and notation

A vector is determined by two pieces of data: its magnitude (the length of the arrow) and its direction and sense. The vector from point $A$ to point $B$ is written $\overrightarrow{AB}$; when the points do not matter, a bold letter is used, such as $\mathbf{u}$ or $\mathbf{v}$. It is drawn as an arrow that starts at $A$ and ends at $B$.

What really characterises a vector is the displacement it makes. It is written as a column of two numbers, the components:

$$\mathbf{u}=\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$$

$u_1$ is the horizontal displacement (positive to the right) and $u_2$ the vertical one (positive upwards).

The figure explains it: to go from $A$ to $B$ you move $x_2-x_1$ horizontally and $y_2-y_1$ vertically. If a value is negative, the displacement is to the left or downwards. For example, for $A(2,5)$ and $B(6,1)$ we have $\overrightarrow{AB}=\left(\begin{array}{c}4\\ -4\end{array}\right)$: four units to the right and four down. The return journey is $\overrightarrow{BA}=\left(\begin{array}{c}-4\\ 4\end{array}\right)$, the opposite vector, written $-\overrightarrow{AB}$.

The same relation works in reverse: if you know $A$ and the components of the vector, the end point is $B=A+\overrightarrow{AB}$, that is, you add the components to the coordinates of $A$.

Equal vectors, zero vector and position vector

Two vectors are equal if they have the same magnitude, the same direction and the same sense. In practice this is all you need:

Two equal arrows can sit in different places in the plane: they are the same displacement drawn from different starting points. That is why we speak of a vector $\mathbf{u}$ without saying where it starts. If $ABCD$ is a parallelogram (with the vertices in that order), opposite sides give equal vectors: $\overrightarrow{AB}=\overrightarrow{DC}$ and $\overrightarrow{AD}=\overrightarrow{BC}$. This lets you find a missing vertex by equating components.

The zero vector $0=\left(\begin{array}{c}0\\ 0\end{array}\right)$ is the displacement that moves nothing ($\overrightarrow{AA}$).

The position vector of a point $P$ is the vector from the origin $O$ to $P$, $\overrightarrow{OP}$. Its components coincide with the coordinates of $P$: if $P(3,-2)$, then $\overrightarrow{OP}=\left(\begin{array}{c}3\\ -2\end{array}\right)$. This is useful because $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$, which is the same as "end minus start".

Magnitude of a vector

The magnitude of $\mathbf{u}$, written $|\mathbf{u}|$, is the length of its arrow. The two components form the legs of a right-angled triangle whose hypotenuse is the arrow, so Pythagoras' theorem applies.

The magnitude is never negative, and it is $0$ only for the zero vector. A vector with $|\mathbf{u}|=1$ is called a unit vector. To obtain one that points in the same direction and sense as $\mathbf{u}\ne 0$, divide by its magnitude; it is usually written $\hat{\mathbf{u}}$:

$$\hat{\mathbf{u}}=\frac{1}{|\mathbf{u}|}\mathbf{u}$$

With $\mathbf{u}=\left(\begin{array}{c}6\\ 8\end{array}\right)$ the magnitude is $\sqrt{36+64}=10$, and the unit vector is $\left(\begin{array}{c}0.6\\ 0.8\end{array}\right)$.

Addition, subtraction and scalar multiplication

These operations are done component by component, and each one has a geometric meaning.

• Addition. It is making one displacement and then the other. If you draw $\mathbf{v}$ starting where $\mathbf{u}$ ends, the arrow from the start to the end is $\mathbf{u}+\mathbf{v}$. It is the diagonal of the parallelogram formed by both vectors drawn from the same point (figure).
• Subtraction. $\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v})$: add the opposite.
• Scalar multiplication. It changes the length without changing the direction: $|k\mathbf{u}|=|k||\mathbf{u}|$. If $k>0$ the sense is that of $\mathbf{u}$; if $k<0$ it is the opposite. With $k=0$ you get the zero vector.

These operations satisfy the usual properties: $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$, $\mathbf{u}+(-\mathbf{u})=0$ and $k(\mathbf{u}+\mathbf{v})=k\mathbf{u}+k\mathbf{v}$. There is also a very useful rule with points:

Parallel vectors

Two vectors are parallel if one is a multiple of the other. That is, they have the same direction (the sense may be opposite).

The form without denominators is preferable because it also works when a component is $0$. If $k>0$ the vectors have the same sense, and if $k<0$, opposite senses.

This lets you check whether three points $A$, $B$ and $C$ are collinear: they are exactly when $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parallel (they share the point $A$, so they lie on the same line). It also lets you find an unknown value: if two vectors are parallel and one of them has a parameter, proportionality gives an equation for that parameter.

Midpoint of a segment

The midpoint $M$ of $AB$ is halfway along, so $\overrightarrow{AM}=\frac{1}{2}\overrightarrow{AB}$. Equating components gives a simple formula.

More generally, the point $P$ on the segment $AB$ with $\overrightarrow{AP}=t\overrightarrow{AB}$ (with $0\le t\le 1$) is $P=A+t\overrightarrow{AB}$. With $t=\frac{1}{2}$ you get the midpoint, and with $t=\frac{1}{3}$ the first of the points that divide the segment into three equal parts.

Base vectors: $\mathbf{i}$ and $\mathbf{j}$

The unit vectors that point along the axes are

$$\mathbf{i}=\left(\begin{array}{c}1\\ 0\end{array}\right)\mathbf{j}=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

Any vector in the plane can be written with them, because its components tell you how many steps to take along each axis:

For example, $\left(\begin{array}{c}3\\ -5\end{array}\right)=3\mathbf{i}-5\mathbf{j}$. In IB problems the two notations are swapped without warning, and the operations are done in the same way: $(2\mathbf{i}+\mathbf{j})+(\mathbf{i}-4\mathbf{j})=3\mathbf{i}-3\mathbf{j}$.