A vector has both magnitude (size) and direction. Vectors are written as column vectors $\begin{pmatrix} x \\ y \end{pmatrix}$ or bold letters (a) or with an arrow ($\overrightarrow{AB}$).
Adding vectors: $$\begin{pmatrix} 3 \\\\ 2 \end{pmatrix} + \begin{pmatrix} -1 \\\\ 4 \end{pmatrix} = \begin{pmatrix} 2 \\\\ 6 \end{pmatrix}$$
Scalar multiplication: multiplying by a scalar scales the magnitude. $$3\begin{pmatrix} 2 \\\\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\\\ -3 \end{pmatrix}$$
Magnitude: $\left|\begin{pmatrix} a \\ b \end{pmatrix}\right| = \sqrt{a^2 + b^2}$
Finding a vector path: $$\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} = -\mathbf{a} + \mathbf{b} = \mathbf{b} - \mathbf{a}$$
Parallel vectors: $\mathbf{p}$ is parallel to $\mathbf{q}$ if $\mathbf{p} = k\mathbf{q}$ for some scalar $k$.
Midpoints: the midpoint $M$ of $AB$ is at $\overrightarrow{OM} = \frac{1}{2}(\mathbf{a} + \mathbf{b})$.
Vector proofs: show two vectors are parallel or that three points are collinear.
Common error: reversing the direction — $\overrightarrow{BA} = -\overrightarrow{AB}$, not $+\overrightarrow{AB}$.
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