Further Maths Homework due 3.2.16

Total: 27 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Find the distance between this line and parallel plane: \[ \underline r = \left(\begin{array}{c} -5\\2\\1\\\end{array}\right) + \lambda \left(\begin{array}{c} -3\\1\\2\\\end{array}\right) \\ \underline r \cdot \left(\begin{array}{c} 1\\3\\0\\\end{array}\right) = 4 \]

   [6 marks]

2. Find the equation of the line where these 2 planes meet in the form $ \underline r \times \underline u = \underline v$ \[x+3y-z=2 \\ 2x-y-z = 1 \]

   [4 marks]

3. Find the eigenvalues and assosciated normalised eigenvectors for this linear transformation: \[ A = \left(\begin{array}{ccc} -2&-4&2 \\ -2&1&2\\ 4&2&5 \\ \end{array}\right) \]

   [8 marks]

4. Find the 3x3 matrix for the transformation represented by T. \[ T: \left(\begin{array}{c} x\\y\\z\\\end{array}\right) \rightarrow \left(\begin{array}{c} x+y\\x-2y\\3z\\\end{array}\right) \] Find the image of the line: \[ \underline r = \left(\begin{array}{c} -5\\2\\1\\\end{array}\right) + \lambda \left(\begin{array}{c} -3\\1\\2\\\end{array}\right) \]

   [4 marks]

5. Find the the shortest distance between these lines \[ \underline r = \left(\begin{array}{c} -1\\1\\0\\\end{array}\right) + \lambda \left(\begin{array}{c} 3\\1\\0\\\end{array}\right) \\ \underline r = \left(\begin{array}{c} -2\\3\\-1\\\end{array}\right) + \mu \left(\begin{array}{c} -3\\4\\2\\\end{array}\right) \]

   [5 marks]