#math/calculus #math/linear-algebra A vector is defined to have a magnitude and direction. The start of the vector is called the `tail` and the end as the `tip` or `head`. The start and end points are known as `initial point` and `terminal point`, respectively. ```tikz \begin{document} \begin{tikzpicture}[line cap=round, line join=round, scale=0.9] % the vector from P (tail) to Q (tip) \draw[->, very thick, blue] (0.5,0.5) -- (4,2.6); % endpoints \fill (0.5,0.5) circle (1.8pt) node[below left]{$P$ \small(tail)}; \fill (4,2.6) circle (1.8pt) node[above right]{$Q$ \small(head)}; % label \node[blue] at (2,2){$\overrightarrow{\mathbf{PQ}}$}; \end{tikzpicture} \end{document} ``` When we are adding vectors together, it's good to observe them as `displacements`. ```tikz \begin{document} \begin{tikzpicture}[line cap=round, line join=round, scale=0.9] % faint axes through the origin \draw[->, black!20] (-0.6,0) -- (5,0); \draw[->, black!20] (0,-1.9) -- (0,3.6); % B redrawn from the tip of A (tip-to-tail) -> reaches A+B \draw[->, dashed, teal!70!black] (3,1) -- (4,3); % other side of the parallelogram \draw[dashed, black!35] (1,2) -- (4,3); % A - B as the displacement from tip of B to tip of A (same as origin vector below) \draw[->, dashed, orange!85!black] (1,2) -- (3,1); % vectors from the origin \draw[->, ultra thick, red!80!black] (0,0) -- (4,3) node[above right]{$\mathbf{A+B}$}; \draw[->, very thick, blue] (0,0) -- (3,1) node[below right]{$\mathbf{A}$}; \draw[->, very thick, teal] (0,0) -- (1,2) node[above left]{$\mathbf{B}$}; \draw[->, very thick, orange!85!black] (0,0) -- (2,-1) node[below right]{$\mathbf{A-B}$}; % origin \fill (0,0) circle (1.6pt); \end{tikzpicture} \end{document} ``` Introducing points, we label the `origin` i.e. $(0,0) \text{ in }\mathbb{R}^{2}$ and $(0,0,0) \text{ in }\mathbb{R}^{3}$ as $O$. When a vector is drawn with its tail on the origin, its referred to as an `origin vector`. Thus with an origin vector, the coordinates of the head determine the vector. For example, using coordinates, we can write an origin vector $\mathbf{A}$ as $ \mathbf{A}=\left\langle{} a_{1},a_{2} \right\rangle = a_{1}\mathbf{i} +a_{2}\mathbf{j} $ The vectors $\mathbf{i}= \left\langle{} 1,0 \right\rangle$ and $\mathbf{j} =\left\langle{} 0,1 \right\rangle$ which represent represent the [[2B1 - Bases|standard basis]]. Addition and subtraction of vectors is as you'd intuit. The `magnitude` of a vector $\mathbf{A}$ is denoted as $|\mathbf{A}|$. If $\mathbf{A}=a_{1}\mathbf{i}+a_{2}\mathbf{j}$, then $|\mathbf{A}|=\sqrt{a_{1}^{2}+a_{2}^{2}}$ . ```tikz \begin{document} \begin{tikzpicture}[line cap=round, line join=round, scale=1.0] % faint axes \draw[->, black!20] (-0.5,0) -- (5.2,0); \draw[->, black!20] (0,-0.6) -- (0,3.7); % A's legs: a1 (horizontal) then a2 (vertical) \draw[very thick, blue] (0,0) -- (3,0) node[midway, below]{$a_1$}; \draw[very thick, blue] (3,0) -- (3,1) node[midway, right]{$a_2$}; % B's legs, starting from the tip of A: b1 then b2 \draw[very thick, teal] (3,1) -- (4,1) node[midway, below]{$b_1$}; \draw[very thick, teal] (4,1) -- (4,3) node[midway, right]{$b_2$}; % the three vectors drawn as the hypotenuses \draw[->, semithick, blue!55!black, dashed] (0,0) -- (3,1) node[pos=0.55, above left]{$\mathbf{A}$}; \draw[->, semithick, teal!55!black, dashed] (3,1) -- (4,3) node[pos=0.5, above left]{$\mathbf{B}$}; \draw[->, ultra thick, red!80!black] (0,0) -- (4,3) node[above]{$\mathbf{A+B}$}; % guides to read the resultant's coordinates (kept off the staircase) \draw[dashed, black!30] (4,1) -- (4,0); \draw[dashed, black!30] (4,3) -- (0,3); \node[red!70!black] at (2,-0.5){$a_1+b_1$}; \node[red!70!black, rotate=90] at (-0.4,1.5){$a_2+b_2$}; \fill (0,0) circle (1.6pt); \end{tikzpicture} \end{document} ``` Note in $\mathbb{R}^{3}$ the magnitude of a vector, $\mathbf{A}$ naturally becomes $|\mathbf{A}|=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$. As a visual, consider $r=\sqrt{a_{1}^{2}+a_{2}^{2}}$ and then $|\mathbf{A}|=\sqrt{r^{2}+a_{3}^{2}}=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$. ```tikz \begin{document} \begin{tikzpicture}[line cap=round, line join=round, scale=1.0] % pseudo-3D axes \draw[->, black!25] (0,0) -- (-1.9,-1.19) node[below left]{$x$}; \draw[->, black!25] (0,0) -- (3.6,0) node[right]{$y$}; \draw[->, black!25] (0,0) -- (0,3.3) node[above]{$z$}; % components a1 (along x) and a2 (along y) forming r in the xy-plane \draw[dashed, black!45] (0,0) -- (-0.88,-0.55) node[midway, above left]{$a_1$}; \draw[dashed, black!45] (-0.88,-0.55) -- (1.5,-0.55) node[midway, below]{$a_2$}; % r : projection of A onto the xy-plane \draw[->, very thick, blue] (0,0) -- (1.5,-0.55) node[below right]{$r$}; % a3 : rise along z \draw[->, very thick, teal] (1.5,-0.55) -- (1.5,1.65) node[midway, right]{$a_3$}; % the vector A and its magnitude \draw[->, ultra thick, red!80!black] (0,0) -- (1.5,1.65) node[above right]{$\mathbf{A},\ |\mathbf{A}|$}; % dashed guide closing the vertical triangle \draw[dashed, black!30] (0,0) -- (1.5,-0.55); \fill (0,0) circle (1.6pt); \end{tikzpicture} \end{document} ``` **Notation** $ \begin{align} (a_{1},a_{2}) & \quad \ldots \quad \text{is a point in the plane} \\ \left\langle{} a_{1},a_{2} \right\rangle = a_{1}\mathbf{i}+a_{2}\mathbf{j} & \quad \ldots \quad \text{is the vector drawn from origin to point } (a_{1},a_{2}) \\ a_{1} \text{ and }a_{2} \text{ in }\mathbf{A}=a_{1}\mathbf{i}+a_{2}\mathbf{j} & \quad \ldots \quad \text{are called the } \mathbf{i } \text{ and }\mathbf{j} \text{ components of }\mathbf{A}\\ \overrightarrow{\mathbf{P}} \text{ and }\overrightarrow{\mathbf{OP}} & \quad \ldots\quad \text{is the origin vector from origin to }\mathbf{P} \\ \overrightarrow{\mathbf{PQ}} & \quad \ldots\quad \text{is the displacement from }\mathbf{P} \text{ to }\mathbf{Q} \end{align} $