Edexcel GCSE Maths: Transformations and Vectors

$OACB$OACB is a parallelogram. $\overrightarrow{OA} = \mathbf{a}$OA=a and $\overrightarrow{OB} = \mathbf{b}$OB=b.

The point $M$M is the midpoint of $AC$AC. The point $N$N lies on $OC$OC such that $ON : NC = 2 : 1$ON:NC=2:1.

(a) Express $\overrightarrow{OM}$OM in terms of $\mathbf{a}$a and $\mathbf{b}$b. (2 marks)

(b) Express $\overrightarrow{MN}$MN in terms of $\mathbf{a}$a and $\mathbf{b}$b, simplifying your answer. (3 marks)

$\mathbf{p} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$p=(3−2​) and $\mathbf{q} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$q=(−14​).

(a) Work out $2\mathbf{p} - \mathbf{q}$2p−q as a column vector. (2 marks)

(b) Work out the magnitude $|\mathbf{q}|$∣q∣, giving your answer correct to 3 significant figures. (2 marks)

Triangle $T$T has vertices at $(2, 1)$(2,1), $(4, 1)$(4,1) and $(2, 4)$(2,4).

Triangle $T$T is enlarged by scale factor $-2$−2 with centre of enlargement $(1, 1)$(1,1) to give triangle $T'$T′.

Work out the coordinates of the three vertices of triangle $T'$T′. (4 marks)

Triangle $A$A is mapped onto triangle $B$B by a reflection in the line $y = x$y=x. Triangle $B$B is then mapped onto triangle $C$C by a reflection in the $x$x-axis.

A single transformation maps triangle $A$A directly onto triangle $C$C. Describe fully this single transformation. (3 marks)

(You may take a test point such as $(4, 1)$(4,1) on $A$A to help you.)

$\mathbf{a} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$a=(5−3​) and $\mathbf{b} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}$b=(26​).

Work out $\mathbf{a} + \mathbf{b}$a+b as a column vector. (2 marks)

Write down the column vector that represents a translation of $4$4 units to the left and $3$3 units up. (1 mark)