8 Vectors and transformation geometry

Learning objectives

8A Scalars and vectorsScalar and vector quantities Vectors will be in two dimensions only 8B Vector notationUnderstand and use vector notation The notation for the vector from O to A, written OA with an arrow above it, and the bold vector a will be used, as will column vectors 8C Directed line segmentsRepresentation of a vector by a directed line segment 8D Parallel, unit and position vectorsParallel vectors, unit vectors and position vectors 8E Vector sums and differencesSum and difference of two vectors 8F Vector magnitudeModulus (magnitude) of a vector 8G Scalar multiplication of vectorsMultiplication of a vector by a scalar 8H Resultant vectorsFind the resultant of two or more vectors 8I Vector geometryApply vector methods to simple geometrical problems The problems may involve colinearity, parallel lines and concurrency 8J TransformationsTransformations of the plane Reflections in any line Rotations about any point Translations Enlargements 8K Combined transformationsCombination of transformations 8L Vectors and matricesMultiplication of a vector by a matrix To include the finding of a matrix for a given transformation of the plane, using the column vectors (1, 0)ᵀ and (0, 1)ᵀ These transformations will be those for which the origin is unchanged

A scalar has magnitude only. A vector has both magnitude and direction; in this syllabus vectors are two-dimensional.

Scalar	Vector
mass, time, temperature	displacement, velocity, force
distance	displacement
speed	velocity

Ask whether changing direction while keeping the numerical size would change the quantity. If yes, the quantity is vector-valued.

A negative scalar is still a scalar. A vector is not identified merely by having two numbers: the numbers must encode directed components.

\overrightarrow{OA}=\mathbf a=\begin{pmatrix}a_x\a_y\end{pmatrix}

The top component is horizontal: positive right, negative left. The bottom component is vertical: positive up, negative down.

\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf b-\mathbf a

Order matters: $\overrightarrow{AB}=-\overrightarrow{BA}$ . Do not swap the two components or omit the direction arrow when naming a vector between points.

A directed line segment shows a vector by its length and arrow direction. Its starting position is irrelevant: equal vectors may be drawn in different places.

A(x_1,y_1),\ B(x_2,y_2)\Rightarrow\overrightarrow{AB}=\begin{pmatrix}x_2-x_1\y_2-y_1\end{pmatrix}

From any chosen start, move by the horizontal component, then the vertical component, and place the arrowhead at the endpoint.

A segment without an arrow has no specified direction. Parallel segments of different lengths are not equal vectors, and reversing an arrow changes the sign.

Type	Test
parallel vectors	one is a scalar multiple of the other
unit vector	magnitude is 1
position vector of $P$	vector $\overrightarrow{OP}$ from the origin

\mathbf b=k\mathbf a\Rightarrow \mathbf a\parallel\mathbf b\quad(k<0\text{ gives opposite directions})

\widehat{\mathbf a}=\frac{\mathbf a}{|\mathbf a|}\quad(\mathbf a\ne\mathbf0)

Parallel does not mean equal: the multiplier may change length or reverse direction. The zero vector has no unit direction and cannot be normalised.

\begin{pmatrix}a\b\end{pmatrix}+\begin{pmatrix}c\d\end{pmatrix}=\begin{pmatrix}a+c\b+d\end{pmatrix}

For addition, place vectors head-to-tail; the resultant joins the first tail to the last head. Subtracting $\mathbf b$ means adding the reversed vector $-\mathbf b$ .

\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}

Only combine matching horizontal and vertical components. Vector subtraction is not commutative, and a valid path must respect each arrow's direction.

\left|\begin{pmatrix}x\y\end{pmatrix}\right|=\sqrt{x^2+y^2}

Magnitude is the non-negative length of the vector. It follows from Pythagoras applied to the horizontal and vertical components.

\left|\begin{pmatrix}-6\8\end{pmatrix}\right|=\sqrt{36+64}=10

Square each signed component before adding and give a non-negative result. $|\mathbf a|$ is a scalar, not a column vector.

k\begin{pmatrix}a\b\end{pmatrix}=\begin{pmatrix}ka\kb\end{pmatrix}

Multiplier $k$	Effect
$k>1$	same direction, longer
$0<k<1$	same direction, shorter
$k<0$	reverse direction, length scaled by $\|k\|$
$k=0$	zero vector

|k\mathbf a|=|k|,|\mathbf a|

Multiply every component. A negative multiplier reverses direction; it does not create a negative magnitude.

The resultant is the single vector with the same overall effect as all the given vectors applied in sequence.

\mathbf r=\mathbf v_1+\mathbf v_2+\cdots+\mathbf v_n

Choose a consistent positive direction, convert every vector to components, add components, then state the result as a directed vector. A closed head-to-tail path has resultant $\mathbf0$ .

Do not add magnitudes unless all vectors lie on the same line in the same direction. Opposing components must carry opposite signs.

Goal	Vector evidence
collinear points	connecting vectors are scalar multiples
parallel lines	direction vectors are scalar multiples
same midpoint	position-vector averages are equal
concurrency	independently derived position vectors give the same point

\overrightarrow{OP}=\overrightarrow{OA}+\lambda\overrightarrow{AB}

Express every route from a common origin, simplify to the same vector basis, and compare coefficients or scalar multiples. State the geometric conclusion after the algebra.

Equal-looking coefficients are meaningful only in a consistent vector basis. A scalar-multiple result proves parallelism; extra point-sharing or position information is needed to conclude collinearity.

Transformation	Complete description
reflection	mirror line
rotation	centre, angle, direction
translation	column vector
enlargement	centre and scale factor

Transform every vertex, preserve vertex order, and join the images. Use perpendicular equal distances for reflections, constant centre-distance for rotations, and centre-to-point rays for enlargements.

\begin{pmatrix}x\y\end{pmatrix}\mapsto\begin{pmatrix}x+a\y+b\end{pmatrix}

A transformation is not fully described by its type alone. For negative enlargement factors the image lies on the opposite ray from the centre; do not infer a centre or mirror line by appearance alone.

Apply transformations in the stated order and use the first image as the input to the second. The order can change the final result.

P\xmapsto{T_1}P'\xmapsto{T_2}P''

Track one labelled vertex at a time, record intermediate coordinates, then transform the remaining vertices consistently. To describe a combined result, compare the original and final shapes only after the chain is complete.

Do not collapse two transformations by guessing from the picture. Rotations, reflections and translations generally do not commute, so reversing the order may produce another image.

\begin{pmatrix}a&b\c&d\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}

The columns are the images of the basis vectors: transform $(1,0)^T$ to get column 1 and $(0,1)^T$ to get column 2.

M\begin{pmatrix}1\0\end{pmatrix}=\begin{pmatrix}a\c\end{pmatrix},\qquad M\begin{pmatrix}0\1\end{pmatrix}=\begin{pmatrix}b\d\end{pmatrix}

A $2 imes2$ matrix here represents only transformations that leave the origin fixed. Ordinary translations cannot be encoded by such a matrix; keep point vectors as $(x,y)^T$ .

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8 Vectors and transformation geometry

Learning objectives

Distinguish scalars from vectors

Read and write vector notation

Represent a vector as a directed segment

Recognise parallel, unit and position vectors

Add and subtract vectors componentwise

Calculate vector magnitude

Scale a vector

Find a resultant from several vectors

Prove geometric relationships with vectors

Describe and perform plane transformations

Combine transformations in sequence

Use matrices for origin-fixed transformations

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