Transformations and vectors flashcards

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What information is needed to describe a reflection?
The equation of the mirror line, e.g. \(x=2\) or \(y=-x\). Each point and its image are the same perpendicular distance from the line, on opposite sides.
Reflect the point \((3,5)\) in the \(x\)-axis, then in the line \(y=x\).
In the \(x\)-axis: \((3,-5)\). Reflecting \((3,-5)\) in \(y=x\) swaps its coordinates: \((-5,3)\).
What information is needed to describe a rotation?
The angle, the direction (clockwise or anticlockwise) and the centre of rotation. A \(180^\circ\) rotation needs no direction.
Rotate \((2,1)\) by \(90^\circ\) anticlockwise about the origin.
\((x,y)\to(-y,x)\), so the image is \((-1,2)\).
How do you find the centre of a rotation from a shape and its image?
Join two pairs of corresponding points and draw the perpendicular bisector of each. The bisectors meet at the centre of rotation.
How do you describe a translation, and how is it written?
As a column vector \(\begin{pmatrix}a\\b\end{pmatrix}\): \(a\) units right (left if negative) and \(b\) units up (down if negative). Every point moves the same way.
What information is needed to describe an enlargement?
The scale factor and the centre of enlargement.
(Extended) A triangle is enlarged with scale factor 3. How do the side lengths and area change?
Side lengths are multiplied by 3 and the area by \(3^2=9\). The angles stay the same.
(Extended) What do a fractional and a negative scale factor do in an enlargement?
A fraction between 0 and 1 makes the image smaller. A negative factor puts the image on the opposite side of the centre and inverted, with lengths multiplied by the size of the factor.
(Extended) How do you find the centre of an enlargement from a shape and its image?
Draw lines through each pair of corresponding vertices. The lines meet at the centre of enlargement. The ratio of distances gives the scale factor.
Which properties stay the same under reflection, rotation and translation, and under enlargement?
Reflection, rotation, translation: size and shape are unchanged (the image is congruent). Enlargement: shape is kept (similar) but size changes.
What is the difference between a vector and a scalar?
A vector has both magnitude and direction, e.g. displacement, written \(\begin{pmatrix}2\\3\end{pmatrix}\) or \(\overrightarrow{AB}\). A scalar has magnitude only, e.g. a number.
(Extended) If \(A=(1,2)\) and \(B=(5,5)\), find \(\overrightarrow{AB}\).
End minus start: \(\overrightarrow{AB}=\begin{pmatrix}5-1\\5-2\end{pmatrix}=\begin{pmatrix}4\\3\end{pmatrix}\).
(Extended) Calculate \(\begin{pmatrix}3\\-2\end{pmatrix}+2\begin{pmatrix}1\\4\end{pmatrix}\).
Multiply then add component by component: \(\begin{pmatrix}3+2\\-2+8\end{pmatrix}=\begin{pmatrix}5\\6\end{pmatrix}\).
(Extended) How do you find the magnitude of a vector \(\begin{pmatrix}a\\b\end{pmatrix}\)?
\(\left|\mathbf{v}\right|=\sqrt{a^2+b^2}\) by Pythagoras. E.g. \(\begin{pmatrix}5\\12\end{pmatrix}\) has magnitude 13.
(Extended) What does \(-\mathbf{a}\) mean compared with \(\mathbf{a}\)?
Same length but opposite direction. Also \(\overrightarrow{BA}=-\overrightarrow{AB}\).
(Extended) What can you say about vectors \(\mathbf{a}\) and \(k\mathbf{a}\)?
They are parallel. \(k\mathbf{a}\) has \(|k|\) times the length, and points the opposite way if \(k\) is negative.
(Extended) How do you show three points \(A\), \(B\), \(C\) lie on a straight line using vectors?
Show \(\overrightarrow{AB}\) is a multiple of \(\overrightarrow{BC}\) (or \(\overrightarrow{AC}\)). Parallel vectors sharing a common point mean the points are collinear.
(Extended) \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\). Write \(\overrightarrow{AB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
Go from \(A\) back to \(O\) then on to \(B\): \(\overrightarrow{AB}=-\mathbf{a}+\mathbf{b}=\mathbf{b}-\mathbf{a}\).
(Extended) \(M\) is the midpoint of \(AB\). Write \(\overrightarrow{OM}\) using \(\mathbf{a}\) and \(\mathbf{b}\).
\(\overrightarrow{OM}=\mathbf{a}+\frac12(\mathbf{b}-\mathbf{a})=\frac12(\mathbf{a}+\mathbf{b})\).
(Extended) \(P\) lies on \(AB\) with \(AP:PB=1:2\). Find \(\overrightarrow{OP}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
\(\overrightarrow{AP}=\frac13\overrightarrow{AB}=\frac13(\mathbf{b}-\mathbf{a})\). So \(\overrightarrow{OP}=\mathbf{a}+\frac13(\mathbf{b}-\mathbf{a})=\frac23\mathbf{a}+\frac13\mathbf{b}\).
Triangle \(A(1,1)\), \(B(3,1)\), \(C(3,4)\) is translated by \(\begin{pmatrix}-2\\3\end{pmatrix}\). Find the image of \(C\).
Add the vector: \((3-2,\ 4+3)=(1,7)\).
Describe the single transformation that maps \((x,y)\) to \((-x,y)\).
A reflection in the \(y\)-axis (the line \(x=0\)).
Triangle with vertices \((1,1)\), \((2,1)\), \((1,3)\) is enlarged by scale factor 2 about the origin. Find the images.
Multiply each coordinate by 2: \((2,2)\), \((4,2)\), \((2,6)\).
(Extended) Enlarge \((4,6)\) with scale factor \(-\frac12\) about the origin.
Multiply coordinates by \(-\frac12\): \((-2,-3)\).
What is the effect of a translation \(\begin{pmatrix}a\\b\end{pmatrix}\) followed by its negative \(\begin{pmatrix}-a\\-b\end{pmatrix}\)?
The shape returns to its original position. The second translation is the inverse of the first.
How do you describe a transformation fully in an exam answer?
Give the single transformation's name plus every detail it needs: reflection (line), rotation (angle, direction, centre), translation (vector), enlargement (scale factor, centre).
(Extended) A \(2\times2\) square is enlarged by scale factor 4. By what factor does the perimeter and area change?
Perimeter is multiplied by 4 (from 8 to 32). Area is multiplied by 16 (from 4 to 64).

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