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(b) the volume of the parallelepiped determined by u, v, and w

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MA 1111: Linear Algebra I

Homework problems for September 21, 2018

Solutions to this problem sheet are to be handed in after our class at 1pm on Friday. Please attach a cover sheet with a declaration

http://tcd-ie.libguides.com/plagiarism/declaration

confirming that you know and understand College rules on plagiarism. On the same cover sheet, please put your name, student number, and name of the degree (Maths/TP/TSM), and staple all the sheets together. (Failure to do that may result in misplaced/lost sheets, for which no responsibility can be taken by instructors.)

1.(a) Does the straight line passing through the points(6, 8) and(9, 13) contain the point (1, 0)? (b) Do the points (1, 1), (4, 5), and (9,−5) form a right triangle?

In the next two questions, we consider the vectors

u= (1,−1, 1), v= (2, 3,−1), w= (0, 2, 1).

2.Compute the following products:

u·v, v·w, v×w, u×w, u·(v×w), v·(u×w).

3. Use your results from the previous question to compute (a) the area of the parallelogram determined by the vectors u and w; (b) the volume of the parallelepiped determined by u, v, and w.

4. Prove that the coordinates of the point (x0, y0) where the [counter- clockwise] rotation through α about the point (0, 0) brings the given point (x, y) are

x0 =xcosα−ysinα, y0 =xsinα+ycosα.

(Hint: show that for the points (x, y) = (1, 0) and (x, y) = (0, 1) directly, and then use the fact that the vector from the origin to the point (x, y) is equal to the vector x·(1, 0) +y·(0, 1).)

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