Vectors and transformation geometry flashcards

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How do you read and write a column vector such as \(\begin{pmatrix}3\\-2\end{pmatrix}\)?
The top number is the movement right (negative means left) and the bottom number is the movement up (negative means down). So this is 3 right and 2 down.
How can a vector be named in writing and in print?
By its end points with an arrow, \(\overrightarrow{AB}\), meaning from A to B, or by a single letter, bold in print (a) and underlined when handwritten.
Work out \(\begin{pmatrix}2\\5\end{pmatrix}+\begin{pmatrix}-4\\1\end{pmatrix}\).
Add top to top and bottom to bottom: \(\begin{pmatrix}-2\\6\end{pmatrix}\).
Work out \(3\begin{pmatrix}2\\-1\end{pmatrix}\) and say how it relates to the original vector.
\(\begin{pmatrix}6\\-3\end{pmatrix}\). A scalar multiple is parallel to the original and 3 times as long.
How are \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) related?
\(\overrightarrow{BA}=-\overrightarrow{AB}\). Reversing direction reverses the sign of every component.
Find the magnitude of \(\begin{pmatrix}5\\12\end{pmatrix}\).
Use Pythagoras: \(\sqrt{5^2+12^2}=\sqrt{169}=13\). Magnitude is a length, so it is never negative.
Find the vector AB for A(1, 4) and B(6, −2).
\(\overrightarrow{AB}=\) end minus start \(=\begin{pmatrix}6-1\\-2-4\end{pmatrix}=\begin{pmatrix}5\\-6\end{pmatrix}\).
If \(\mathbf a=\begin{pmatrix}1\\2\end{pmatrix}\) and \(\mathbf b=\begin{pmatrix}3\\-1\end{pmatrix}\), find \(2\mathbf a-\mathbf b\) and its magnitude.
\(2\mathbf a-\mathbf b=\begin{pmatrix}2\\4\end{pmatrix}-\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}-1\\5\end{pmatrix}\). Magnitude \(=\sqrt{1+25}=\sqrt{26}=5.10\) (3 s.f.).
How can you tell if two vectors are parallel?
One is a scalar multiple of the other. For example \(\begin{pmatrix}2\\3\end{pmatrix}\) and \(\begin{pmatrix}-4\\-6\end{pmatrix}\) are parallel since the second is \(-2\) times the first.
In parallelogram OABC with \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\), find \(\overrightarrow{OB}\) and \(\overrightarrow{AC}\).
\(\overrightarrow{OB}=\mathbf a+\mathbf c\) (opposite sides are equal and parallel). \(\overrightarrow{AC}=\overrightarrow{AO}+\overrightarrow{OC}=\mathbf c-\mathbf a\).
(Higher) M is the midpoint of AB, with \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=\mathbf b\). Find \(\overrightarrow{OM}\).
\(\overrightarrow{AB}=\mathbf b-\mathbf a\), so \(\overrightarrow{OM}=\mathbf a+\tfrac12(\mathbf b-\mathbf a)=\tfrac12(\mathbf a+\mathbf b)\).
(Higher) P lies on AB with AP : PB = 2 : 3. Find \(\overrightarrow{OP}\).
\(\overrightarrow{AP}=\tfrac25\overrightarrow{AB}\), so \(\overrightarrow{OP}=\mathbf a+\tfrac25(\mathbf b-\mathbf a)=\tfrac15(3\mathbf a+2\mathbf b)\).
(Higher) How do you prove that three points A, B, C are collinear using vectors?
Show \(\overrightarrow{AB}=k\overrightarrow{BC}\) for some number \(k\). The vectors are parallel and share the point B, so A, B, C lie on one straight line.
Common error: what goes wrong when finding a vector by a route through other points?
Going against the direction of a known vector needs a minus sign. For example going from B back to A uses \(-\overrightarrow{AB}\). Write the route first, then substitute.
What information fully describes a translation?
Only a column vector, e.g. \(\begin{pmatrix}-3\\4\end{pmatrix}\). Every point moves the same distance in the same direction; the shape's size and orientation are unchanged.
A vertex at \((2,-1)\) is translated by \(\begin{pmatrix}-3\\4\end{pmatrix}\). Where does it go?
Add the vector: \((2-3,\,-1+4)=(-1,3)\).
What must you state to describe a reflection fully?
The equation of the mirror line, for example \(x=2\), \(y=-1\), \(y=x\) or \(y=-x\). Each point and its image are the same distance from the mirror line.
Reflect the point (3, 1) in the y-axis, in y = x, and in y = −x.
Y-axis: \((-3,1)\). Line \(y=x\): swap coordinates, \((1,3)\). Line \(y=-x\): swap and change both signs, \((-1,-3)\).
What must you state to describe a rotation fully?
The centre of rotation, the angle, and the direction (clockwise or anticlockwise). For 180° the direction is not needed.
Rotate (3, 1) by 90° clockwise and by 90° anticlockwise about the origin.
Clockwise: \((x,y)\to(y,-x)\), giving \((1,-3)\). Anticlockwise: \((x,y)\to(-y,x)\), giving \((-1,3)\). Check with a quick sketch.
What must you state to describe an enlargement fully?
The scale factor and the centre of enlargement. Lengths are multiplied by the scale factor; angles stay the same, so the image is similar to the object.
Enlarge P(4, 5) by scale factor 2 about the centre (1, 1).
The vector from the centre to P is \((3,4)\). Double it to \((6,8)\) and add to the centre: \((7,9)\).
(Higher) What happens in an enlargement with a negative scale factor? Enlarge (3, 1) by −2 about the origin.
The image is on the opposite side of the centre and is inverted (rotated 180°), with lengths multiplied by 2. \((3,1)\to(-6,-2)\).
(Higher) What does a fractional scale factor such as ½ do?
The image is smaller than the object, but it is still an enlargement. A scale factor between 0 and 1 reduces lengths; the image is on the same side of the centre as the object.
How do you find the centre and scale factor of an enlargement from a diagram?
Draw lines through corresponding corners of object and image; they meet at the centre. The scale factor is an image length divided by the matching object length.
Which transformations give congruent shapes and which do not?
Translations, reflections and rotations keep size and shape, so the image is congruent. An enlargement (scale factor not 1 or −1) gives a similar, not congruent, shape.
What is an invariant point?
A point that does not move under the transformation. Points on the mirror line are invariant under reflection; the centre is invariant under a rotation or an enlargement.
(Higher) What single transformation equals two reflections in parallel lines, or in intersecting lines?
Parallel mirror lines: a translation, perpendicular to the lines, of twice the distance between them. Intersecting lines: a rotation about their intersection, through twice the angle between them.

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