3.10 Vectors: concepts and algebraIB Maths: Applications and Interpretation HL: Revision notes

A scalar has magnitude (size) only, such as mass, time or speed. A vector has magnitude and direction, such as displacement, velocity or force.

A vector is drawn as a directed line segment: an arrow whose length shows the magnitude and whose arrowhead shows the direction. The vector from $A$ to $B$ is written $\overrightarrow{AB}$, or as a single bold letter $\mathbf{v}$ (handwritten as $\underline{v}$).

Two vectors are equal if they have the same magnitude and direction, wherever they are drawn. The vector $-\mathbf{v}$ has the same magnitude as $\mathbf{v}$ but the opposite direction, and $\overrightarrow{BA}=-\overrightarrow{AB}$. The zero vector $\mathbf{0}$ has no magnitude and no direction.

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 terms of its components: $\mathbf{v}=\begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}.$ Column form and $\mathbf{i},\mathbf{j},\mathbf{k}$ form are interchangeable. In two dimensions use $\mathbf{v}=v_1\mathbf{i}+v_2\mathbf{j}$.

Example: $\begin{pmatrix}2 \\ -1 \\ 2\end{pmatrix}=2\mathbf{i}-\mathbf{j}+2\mathbf{k}$. The zero vector is $\begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix}$ and $-\mathbf{v}$ has every component of opposite sign.

Add or subtract vectors component by component: $\begin{pmatrix}a_1 \\ a_2 \\ a_3\end{pmatrix}\pm\begin{pmatrix}b_1 \\ b_2 \\ b_3\end{pmatrix}=\begin{pmatrix}a_1\pm b_1 \\ a_2\pm b_2 \\ a_3\pm b_3\end{pmatrix}.$ Geometrically, to add, place the tail of the second vector at the head of the first (the triangle rule); the sum goes from the first tail to the last head. To subtract, add the negative.

Multiplying by a scalar $k$ multiplies every component: $k\mathbf{v}$ has magnitude $|k|\,|\mathbf{v}|$, and the same direction if $k>0$ or the opposite if $k<0$. Two vectors are parallel if one is a scalar multiple of the other: $\mathbf{a}=k\mathbf{b}$.

The resultant of several vectors, such as forces acting on a body, is their sum. Example: $\mathbf{F}_1=6\mathbf{i}+8\mathbf{j}$ and $\mathbf{F}_2=-2\mathbf{i}+5\mathbf{j}$ have resultant $4\mathbf{i}+13\mathbf{j}$.

The magnitude of $\mathbf{v}=\begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}$ is $|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}.$ A unit vector has magnitude 1. To normalise a vector, divide it by its magnitude: $\frac{\mathbf{v}}{|\mathbf{v}|}$ is the unit vector in the direction of $\mathbf{v}$. To rescale it to magnitude $m$, multiply the unit vector by $m$: $\frac{m}{|\mathbf{v}|}\mathbf{v}$.

Example: find the velocity of a particle with speed $7$ m s$^{-1}$ in the direction $3\mathbf{i}+4\mathbf{j}$. $|3\mathbf{i}+4\mathbf{j}|=5$, so $\text{velocity}=\frac75(3\mathbf{i}+4\mathbf{j})=4.2\mathbf{i}+5.6\mathbf{j}\ \text{m s}^{-1}.$ Check: $\sqrt{4.2^2+5.6^2}=7$.

The position vector of a point $A$ relative to the origin $O$ is $\overrightarrow{OA}=\mathbf{a}$. The vector from $A$ to $B$ is $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a},$ "end minus start". The distance $AB$ is $|\overrightarrow{AB}|$. The midpoint $M$ of $[AB]$ has position vector $\frac12(\mathbf{a}+\mathbf{b})$.

Example: $A(1,4,-2)$ and $B(3,7,4)$ give $\overrightarrow{AB}=\begin{pmatrix}2 \\ 3 \\ 6\end{pmatrix}$, $AB=\sqrt{4+9+36}=7$, and $M=(2,5.5,1)$.

For a body moving with constant velocity $\mathbf{v}$ from the origin, its position after time $t$ is $t\mathbf{v}$.