IB Mathematics Applications & Interpretation Topic…

In a game animation, each vertex with position vector $\mathbf{r}$ is updated by $T:\mathbf{r}\mapsto\mathbf{r}'$ , where $(\begin{array}{c} x^{'} \\ y^{'} \end{array}) = ($ .

(a) the image of the point $(-4,3)$ under $T$ .

(b) the coordinates $(u,v)$ of the point whose image is twice its position vector.

(e) $\boldsymbol{T}^2$ .

(f) Hence what the value of $\boldsymbol{T}^2$ indicates about the final position of the outline if the complete sequence of operations is performed twice.

(g) Consider three points $P, Q$ , and $R$ on the original outline. After the software applies the transformations, these points move to $P', Q'$ , and respectively.

(h) the specific geometric transformation that corresponds to matrix $\boldsymbol{T}$ .

In a robotics simulation, each waypoint with position vector $\mathbf{r}$ is updated by $S:\mathbf{r}\mapsto\mathbf{r}'$ , where $\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}1&2\\-1&-1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}2\\-3\end{pmatrix}$ .

A geographic information system stores the outline of a lake as coordinate vectors in the $x$ - $y$ plane relative to a reference point $O$ . To compare two survey layers, the software applies the following three operations to every point on the outline:

an anticlockwise rotation about $O$ through $\frac{\pi}{2}$ radians
a mirror reflection across the $y$ -axis
a clockwise rotation about $O$ through $\frac{\pi}{4}$ radians. The software carries out the operations in the order listed.

In a robotics drawing program, each point with position vector $\mathbf{r}$ is moved by $S:\mathbf{r}\mapsto\mathbf{r}'$ , where $\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}2&1\\-1&0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}1\\4\end{pmatrix}$ .

A line passes through the point $A(1,-2)$ with direction vector $\mathbf{b}=\begin{pmatrix}3\\4\end{pmatrix}$ .

(a) Write down the vector equation of the line in the form $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$ , where $\lambda$ is a scalar parameter.

(b) Find the parametric equations of the line.

(a) the image of the point $(3,-2)$ under $S$ .

(b) the coordinates $(u,v)$ of the point whose image is twice its position vector.

(a) the matrix for an anticlockwise rotation about $O$ through $\frac{\pi}{2}$ radians.

(b) the matrix for a mirror reflection across the $y$ -axis.

(d) the composite matrix $\boldsymbol{T}$ that represents the resulting mapping by finding the product of these three matrices in the appropriate order.

(a) the image of the point $(-2,3)$ under $S$ .

(b) the coordinates $(p,q)$ of the point whose image is twice its position vector.

IB Mathematics Applications & Interpretation AHL 3.9—matrix Transformations Questionbank

IB Mathematics Applications & Interpretation AHL 3.9—matrix Transformations Questionbank

IB Mathematics Applications & Interpretation AHL 3.9—matrix Transformations Questionbank

IB Mathematics Applications & Interpretation AHL 3.9—matrix Transformations Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 3.9—matrix Transformations Questionbank