The shape BCDE (in blue) is a rectangle whose centre is at A(-8.5, 4.2) (in black). It is given that OA || BC, |BC| = 7.6 and |BE| = 5.4. For the following, enter your answers in the form (x, y) for both vectors and points. For example, if your answer is (-1.23, 4.56), enter (-1.23, 4.56) in the answer box.

Area =_____. Find the area of the region enclosed between f(x) = 0.6 x^2 + 2, g(x) = x, x = -4, and x = 4

(a) Determine the transformation matrix describing a reflection at the axis y = -x+3 in homogenous coordinates. (b) Given are the three points \mathbf{x}_{1}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right), \mathbf{x}_{2}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \mathbf{x}_{3}=\left(\begin{array}{l} 1 \\ 2 \end{array}\right) as well as their images after transformation \mathbf{x}_{1}^{\prime}=\left(\begin{array}{c} -1 \\ 5 \end{array}\right), \mathbf{x}_{2}^{\prime}=\left(\begin{array}{c} -1 \\ 1 \end{array}\right), \mathbf{x}_{3}^{\prime}=\left(\begin{array}{l} 1 \\ 5 \end{array}\right) Determine the affine transformation matrix T in homogenous coordinates such that x = Tx; for i = 1,2,3! (c) Determine the transformation matrix T in homogenous coordinates for the following transformation: \left(\begin{array}{l} x \\ y \end{array}\right) \longmapsto\left(\begin{array}{c} \cos ^{2} \alpha x-\sin ^{2} \alpha y+\frac{\pi}{2} \\ \tan ^{2} \alpha y+\cos ^{2} \alpha x-\pi \end{array}\right) (d) Determine the transformation matrix T in homogenous coordinates for the following transformation: \left(\begin{array}{l} x \\ y \end{array}\right) \longmapsto\left(\begin{array}{c} \pi \\ \frac{3 y}{x+\pi}+2 \end{array}\right)

\text { 1. Let } \vec{u}=(-1,0,2), \vec{v}=(1,2,3), \vec{w}=(0,-2,1) \text {. Find: } \text { (a) }(\|\vec{v}\| \vec{w}) \cdot(\|\vec{w}\| \vec{v}) \text { (b) } \vec{u} \cdot(\vec{w} \times \vec{v}) \text { (c) } \vec{u} \times(\vec{v} \times \vec{w}) \text { (d) } \operatorname{proj}_{\vec{v}} \vec{u}

. Suppose a particle moves according to the position function r(t) = (cos(1), sin(t), t).Write explicitly the acceleration function and its decomposition into tangential and normal components -i.e. compute explicitly a(t), aī, T(t), an, N(t) in the expression: a(t)ar T(t) + anN(t).

\text { Let } q \in(0,1) \text {. Prove that } \nu=\sum_{k=0}^{\infty}(1-q) q^{k} \delta_{k} \text { is a probability measure on }(\mathbb{R}, \mathscr{B}(\mathbb{R})) \text {. Here } \delta_{k} \text { denotes the Dirac measure } with mass at k. (ii) Let q = 1/2 in the definition of v above. Find: (а) v([-1,1); (b) v((0, 1)); v(N), where N = {1, 2, 3, . .}.

Express the following complex numbers in the rectangular (x+ jy} form: \text { (a) } \quad(1+j)^{2}+(1+j)^{4} \text { (b) } \frac{(1-j \sqrt{3})^{2}}{(j-1)^{38}} \text { (c) }\left(\frac{2 j}{(j+\sqrt{3})}\right)^{19}

Evaluate the following magnitudes: \text { a) }\left|\frac{e^{j \pi}}{1+j}\right| \text { b) }\left|\frac{1+j}{1-j}\right| \text { c) }\left|\frac{\sqrt{5}+3 j}{1-j}\right|

A certain plane is described by 2x + 3y + 4z = 16. Find the unit vector normal to the surface in the direction away from the origin.

Find the curl of the vector field v = (a · r) a × r, where a is a constant vector and r = (x, y, z)is the position vector.