MATH 51TA Section Notes for Thu 25 Sep 08Jason LoToday’s topics:• spans• parametric representations of lines and planes (in R3)Definition (span) Given a (nonempty) set of vectors, their span is the set ofall their linear combinations. That is,span(v1, · · · , vk) := {c1v1+ · · · ckvk: ci∈ R for 1 ≤ i ≤ k}Example 1. What is the geometric description of span010,101?Ans. The plane through the points (0, 1, 0), (1, 0, 1) and the origin (0, 0, 0).Example 2. Give a parametric representation of the line through (0, 2, 0) andthe origin.Ans. In general, a parametric representation of a line in Rnis a set of theform{x0+ tv : t ∈ R}(this is the line through the point x0, parallel to v ). We need 1 reference pointand 1 vector to specify the direction.For this example, one point on the line would be the origin (0, 0, 0). And avector parallel to the line would be020. So a parametric representation of itwould be{0 + t020: t ∈ R} = {t020: t ∈ R}Example 3. Give a parametric representation of the plane in R3containingthe two lines{010+ s10−1: s ∈ R} and {010+ t002: t ∈ R}Ans. These two lines are given by parametric representations. Before webegin, recall that, in general, parametric representations of planes in Rnaresets of the form{x0+ sv1+ tv2: s, t ∈ R}1(this is the plane through the point x0, and containing the vectors v1, v2).You can think of this as the result of translating the plane span(v1, v2) by thevector x0. Note that you need 1 reference point and 2 vectors to specify the”direction”.Back to the problem at hand. We obvious have the point (0, 1, 0) on theplane. Now we need 2 vectors to specify the direction. (Draw the two lines.)We can take the two vectors10−1,002, which are contained in the plane andnon-collinear. So a parametric representation of this plane is010+ s10−1+ t002: s, t ∈ R ⊂ R3Example 4. Parametrise the plane in R3defined by{(x, y, z) ∈ R3: x − y − z = 0}Ans. We need to find 1 reference point and two vectors contained in theplane. This can be achieved by finding three (non-collinear) points on the planefirst. (What happens if the three points are collinear?)Example 5. Find a parametrisation of the plane in R3containing the points(1, 0, 0), (0, 1, 0) and the vector222.Ans. We need to find 1 reference point and two vectors contained in theplane. How do we produce another
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