ベクトル

A column vector $\left(\begin{array}{c}x\\ y\end{array}\right)$ describes a movement of $x$ in the $x$-direction and $y$ in the $y$-direction. The top number is the movement to the right (left if negative) and the bottom number is the movement up (down if negative). For example, $\left(\begin{array}{c}3\\ -2\end{array}\right)$ means '$3$ right and $2$ down'. A vector is written as a bold lower-case letter $\mathbf{a}$, or as $\overrightarrow{AB}$ from a start point to an end point.

To add or subtract two column vectors, work with the top components together and the bottom components together, separately. Addition represents doing one movement followed by the other. For example, $\left(\begin{array}{c}3\\ -1\end{array}\right)+\left(\begin{array}{c}2\\ 4\end{array}\right)=\left(\begin{array}{c}5\\ 3\end{array}\right)$, and subtraction gives $\left(\begin{array}{c}3\\ -1\end{array}\right)-\left(\begin{array}{c}2\\ 4\end{array}\right)=\left(\begin{array}{c}1\\ -5\end{array}\right)$.

To multiply a vector by a number (a scalar), multiply each component by that number: $k\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}kx\\ ky\end{array}\right)$. If $k>0$ the direction is unchanged and the length is multiplied by $k$; if $k<0$ the direction is reversed. For example, $2\left(\begin{array}{c}3\\ -1\end{array}\right)=\left(\begin{array}{c}6\\ -2\end{array}\right)$. If one vector is a scalar multiple of another, the two vectors are parallel.

The magnitude (length) of a vector is found using Pythagoras' theorem. For $\mathbf{a}=\left(\begin{array}{c}x\\ y\end{array}\right)$, the magnitude is $|\mathbf{a}|=\sqrt{x^2+y^2}$. Square each component, add, then take the square root. The signs disappear when you square, so negative components cause no problem. For example, the magnitude of $\left(\begin{array}{c}6\\ -8\end{array}\right)$ is $\sqrt{6^2+(-8{)}^2}=\sqrt{100}=10$.

In geometry problems, the position of a point is described by a vector from a base point (often the origin $O$). If $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$, then the vector from $A$ to $B$ is $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a}$ ('destination minus start'). Midpoints and other points can also be written in terms of $\mathbf{a}$ and $\mathbf{b}$. For example, the midpoint $M$ of $AB$ has $\overrightarrow{OM}=\mathbf{a}+\frac{1}{2}(\mathbf{b}-\mathbf{a})=\frac{1}{2}(\mathbf{a}+\mathbf{b})$.