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Unformatted text preview: MAT1341A: LECTURE NOTES FOR FALL 2008 MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP) 3. Vector Geometry, Part II Last time we discussed the algebra and geometry of vectors in R 2 and R 3 , and extended all the notions of vector addition, scalar multiplication, the dot product, angles and orthogonality to R n . Today let’s consider lines and planes in R 2 and R 3 , following [N, 3.3,3.5]. We’ll see that while some ideas generalize easily to R n , others take more work; in fact, working out what, exactly, the reasonable analogues to “lines and planes” in R n are, is one of our goals in this course. 3.1. Describing Lines. A line in R 2 or R 3 is completely determined by specifying its di rection and a point on the line. Since we prefer working with vectors, replace “point” by its position vector. (Insert picture here) So a line L going through the tip of ~v and such that ~ d is a vector parallel to the direction of L can be described as the set L = { ~v + t ~ d  t ∈ R } . That is, any point ~v on the line L can be written as ~v = ~v + t ~ d for some t . Example: Consider the line y = 3 x + 2 in R 2 . We can get a parametric equation for this line by letting x = t be the parameter and solving for y ; this gives x = t and y = 3 t + 2. In vector form, this is x y = t 3 t + 2 = 0 + 1 t 2 + 3 t = 2 + t 1 3 so the line is L = { 2 + t 1 3  t ∈ R } . Note: Another way to get a parametric equation for this line: it goes through the points (0 , 2) and ( 2 3 , 0), for example. So a direction vector is ~ d = 2 2 / 3 = 2 / 3 2 . Date : September 11, 2008. 1 2 MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP) We can take ~v = (0 , 2). So we get L = 2 + t 2 / 3 2  t ∈ R . NOTICE that our answer is not unique! It depends on our choices. BOTH of our answers are completely correct (check with a sketch!). CAUTION!! We often use “ t ” for the parameter. But if you are comparing two different lines, you must use different letters to represent parameters on different lines! Example: Find the point of intersection of L = { t (1 , 2)  t ∈ R } and L = { (0 , 1) + t (3 , 0)  t ∈ R } . WRONG METHOD: “Set t (1 , 2) = (0 , 1) + t (3 , 0) and solve for t .” This doesn’t give an answer for t ; but that’s only because the two lines don’t arrive at the point of intersection at the same “time” t . The lines still intersect! CORRECT METHOD: Find parameters s and t such that t (1 , 2) = (0 , 1) + s (3 , 0). This gives t = 1 2 and s = 1 6 , and the point of intersection is thus ( 1 2 , 1). The form L = { ~v + t ~ d  t ∈ R } for a line in R n is called the vector form or parametric form . In R 2 (but NOT R 3 ), you can describe a line by an equation like ax + by = c ; this is called normal form ....
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 Winter '10
 DAIGLE
 Algebra, Vectors, Dot Product, Parametric equation, normal vector, Monica Nevins

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