Transformations, congruence, vectors and similarity Study Guide

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Vectors describe displacement with magnitude and direction. Add vectors by combining components or following consecutive journeys, subtract by adding the opposite vector, and multiply by scalars component by component. In geometric proofs, express routes between points in terms of a small set of base vectors.

If a = (3, −2) and b = (−1, 5), then a + b = (2, 3), 2a = (6, −4), and a − b = (4, −7). Parallel vectors are scalar multiples. Points are collinear when the vectors along the same line are scalar multiples with a common point.

A strong solution states the relevant rule, substitutes or transforms values carefully, shows intermediate reasoning, and checks that the result is reasonable and in the required form. Common errors come from omitted terms, mishandled signs, premature rounding or using a rule that does not match the structure of the problem.

Column vectors are a way to describe a translation, which is a movement in a straight line without any rotation or change in size. They tell you how many units to move horizontally and how many units to move vertically. A column vector is typically written as two numbers stacked on top of each other, enclosed in brackets, for example, $\begin{pmatrix} x \\ y \end{pmatrix}$. The top number, $x$, indicates the horizontal movement: a positive value means moving right, and a negative value means moving left. The bottom number, $y$, indicates the vertical movement: a positive value means moving up, and a negative value means moving down.

Adding and subtracting column vectors is straightforward. To add two vectors, you add their corresponding components. For instance, $\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}$. Similarly, to subtract vectors, you subtract their corresponding components: $\begin{pmatrix} a \\ b \end{pmatrix} - \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a-c \\ b-d \end{pmatrix}$. Think of it as performing one translation followed by another, or finding the difference between two translations.

Multiplying a column vector by a scalar (a single number) scales its components. For example, $k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}$. If $k$ is positive, the direction stays the same, but the magnitude of the translation changes. If $k$ is negative, the direction reverses. This means a vector $2\mathbf{a}$ is twice as long and in the same direction as $\mathbf{a}$, while $-\mathbf{a}$ is the same length but in the opposite direction.