1. LAHW#03 Due October 11, 2010

2. 2.1 Euclidean Vector Spaces 13. –Devise a test to determine wherther two lines in R n are the same. Let the lines be described as L 1 = {p + tq | t ∈ R} and L 2 = {v + tw | t ∈ R}. Use your test on this special case: p = (4,2,1), q = (-1,3,2), v = (1,11,7), and w = (3,-9,-6).

3. 2.1 Euclidean Vector Spaces 18. –Show that if u, v, and w are three points in R 2, then for suitable real numbers α, β, and γ, not all zero, we have α u +β v +γ w = 0.

4. 2.2 Lines, Planes, and Hyperplanes 1. –Find a parametric form for the line in R 2 that passes through the points (7, 3) and (-5, 6). Is the answer unique?

5. 2.2 Lines, Planes, and Hyperplanes 2. –In R 5 does the line described parametrically by (3,4,-5,6,2)+ t (2,-2,1,3,6) intersect the line represented by (17,-10,2,27,44)+ t (-3,2,-5,1,4)?

6. 2.2 Lines, Planes, and Hyperplanes 8. –Let P be the set of all vectors X = ( x 1, x 2, x 3, x 4 ) such that

7. 2.2 Lines, Planes, and Hyperplanes 10. –Is there a plane in R 3 that contains the two lines described parametrically by (1,-2,3) + t (1,0,0) and (-2,5,-7) + s (4,-7,10)?

8. 2.2 Lines, Planes, and Hyperplanes 24. –Establish this assertion or find a counterexample: For two lines in R n given parametrically by v + tw and x + sy to intersect, it is necessary and sufficient that x - v be in the span of {w, y}.

9. 2.2 Lines, Planes, and Hyperplanes 25. –Establish this assertion or find a counterexample: A necessary and sufficient condition for the line given parametrically by tu + (1 - t ) v to contain the point 0 is that v be a scalar multiple of u – v.