3.12 Vectors: concepts and algebraIB Maths: Analysis and Approaches HL: Revision notes

A vector has both magnitude (size) and direction; a scalar has only magnitude. A vector can be shown as a directed line segment $\overrightarrow{AB}$, from $A$ to $B$. Two vectors are equal if they have the same magnitude and direction, wherever they are drawn.

• A position vector $\overrightarrow{OA}=\mathbf{a}$ gives the position of $A$ relative to the origin $O$.
• A displacement vector $\overrightarrow{AB}$ describes the move from $A$ to $B$: $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$.

In three dimensions the base vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are unit vectors along the $x$-, $y$- and $z$-axes. Any vector can be written in component form $\mathbf{v}=\begin{pmatrix}v_1\\ v_2\\ v_3\end{pmatrix}=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}.$ The point $A(2,-1,3)$ has position vector $\mathbf{a}=2\mathbf{i}-\mathbf{j}+3\mathbf{k}$.

Work component by component. Geometrically, $\mathbf{u}+\mathbf{v}$ is "$\mathbf{u}$ then $\mathbf{v}$" (the triangle law), and $\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v})$.

• The zero vector $\mathbf{0}$ has every component 0; $\mathbf{v}+(-\mathbf{v})=\mathbf{0}$.
• $-\mathbf{v}$ has the same magnitude as $\mathbf{v}$ but the opposite direction.
• $k\mathbf{v}$ is parallel to $\mathbf{v}$, with magnitude $|k||\mathbf{v}|$; it points the opposite way if $k<0$.

Two non-zero vectors are parallel exactly when one is a scalar multiple of the other. For example $(m,2,n)$ is parallel to $(2,-1,2)$ only if it equals $-2(2,-1,2)$.

The magnitude of $\mathbf{v}$ is $|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$. A unit vector has magnitude 1; the unit vector in the direction of $\mathbf{v}$ is $\frac{\mathbf{v}}{|\mathbf{v}|}$.

The distance between $A$ and $B$ is $|\overrightarrow{AB}|=|\mathbf{b}-\mathbf{a}|$. For $A(2,-1,3)$ and $B(5,3,3)$: $\overrightarrow{AB}=(3,4,0)$ and $AB=5$.

To get a vector of magnitude 6 in the direction of $\mathbf{p}$, use $6\frac{\mathbf{p}}{|\mathbf{p}|}$.

Useful facts:

• The midpoint of $PQ$ has position vector $\frac12(\mathbf{p}+\mathbf{q})$.
• A point dividing $AB$ in the ratio $m:n$ has position vector $\mathbf{a}+\frac{m}{m+n}(\mathbf{b}-\mathbf{a})$.
• $\overrightarrow{KL}=\overrightarrow{NM}$ means $KL$ and $NM$ are parallel and equal, so $KLMN$ is a parallelogram.
• Points $O$, $D$, $F$ are collinear if $\overrightarrow{OF}=k\overrightarrow{OD}$: parallel vectors sharing a point.

Example: joining the midpoints of the sides of any quadrilateral gives a parallelogram, because both $\overrightarrow{KL}$ and $\overrightarrow{NM}$ equal $\frac12(\mathbf{r}-\mathbf{p})$.