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.

Find eigenvalues and eigenvectors for the matrix \left[\begin{array}{cc} 16 & 5 \\ -30 & -9 \end{array}\right] The smaller eigenvalue _________has an eigenvector [ __ ___ ] The larger eigenvalue _____ has an eigenvector[ _____ ______ ]

\text { If } z=x+j y \text { where } x \text { and } y \text { are real calculate the locus of all points such that } \frac{|z-2|}{|z+2|}=2

\text { 2i Find the work done by } \bar{F} \text { along the curve } \vec{r}(t) \text { (c) } \bar{F}(x, y)=\left\langle x^{3} y, x-y\right\rangle, \vec{r}(t) \text { is the parabola } y=x^{2} \text { from }(-2,4) \text { to }(1,1) \text {, }

Question 2.Find the volume of the parallele piped with one vertex at P = (1, 1, 1) and determined by the line segments to Q1 = (1, 2, 3), Q2 = (2, 3, 1), and Q3 = (3, 1, 2).

\text { Suppose that } T: \mathbf{R}^{2} \rightarrow \mathbf{R}^{3} \text { is c inear fransformation and } T(1,1) = (4, 7, 13)T(-2,1) = (1, 1, 2) Find the standard matrix for T and use it to calculate T(-1,2).

Question 5: (1 point) The vertices of a triangle ABC are given by position vectors a, b and c. where a=6i+4j+6k, b=3i+5j+6k, c=7i+6j+5k. Find the area of the triangle ABC giving your answer accurate to four significant figures.

Is w in the subspace spanned by (V₁, V₂, V₁]? Explain why? V1= [10-1], v2=[213], v3=[426] and w= [312]

\text { If } z=x+j y \text { where } x \text { and } y \text { are real calculate the locus of all points such that } \frac{|z-2|}{|z+2|}=2

3. [4 points] Let X be a vector field on Rm such that for every p = Rm the maximal integral curve of X going through p at time zero is defined on the whole of R. Consider the stereographic projection : Sm - {N} → Rm, where N denotes the North Pole. Let Y be a vector field on the sphere Sm such that its restriction to Sm{N} equals (0-¹),X. What can you say about YN, the value of Y at the North Pole?