Class Notes (836,135)
Canada (509,645)
Mathematics (181)
MATH 125 (100)
Scott (32)
Lecture 3

MATH 125 Lecture 3: Lecture 3.PDF

8 Pages
33 Views
Unlock Document

Department
Mathematics
Course
MATH 125
Professor
Scott
Semester
Fall

Description
Recap o Vector addition: If u u1, upl and U v1, v2, then e.g., 1,3 2,4 3, 7 Or Vector negation: If v1, v2, then v2 Vector negation Let U Iv1 vol be a vector in R2. We define its negative, denoted -i, by setting (-1)U v1 V2 Visually, this simply amounts to a reversal of direction. Remark: Note that for any E R2, we have This justifies our notation. Vector subtraction: Given vectors u u1, u2 and U v1, v2 in R2, we define their difference, denoted u v, by setting (-i) In other words, 1, 2 Visually: or Recap o Vector addition: If u u1, upl and U v1, v2, then e.g., 1,3 2,4 3, 7 Or Vector negation: If v1, v2, then v2 Vector negation Let U Iv1 vol be a vector in R2. We define its negative, denoted -i, by setting (-1)U v1 V2 Visually, this simply amounts to a reversal of direction. Remark: Note that for any E R2, we have This justifies our notation. Vector subtraction: Given vectors u u1, u2 and U v1, v2 in R2, we define their difference, denoted u v, by setting (-i) In other words, 1, 2 Visually: orExample 1.6: If 2, 11 and U 3, 2, compute v and draw it in the ane. Solution: W- V C2- (-3), 4-2 Cs,-11 (NB Remark: Note that for any points A and B in the ay-plane, we have This is equivalent to the earlier formula OB OA+A AB OA Example 1.7: If the vector 1,3 is drawn in the zy-plane with initial point A (-2,-1), what is its terminal point? Solution: OA AB OB OB AB OA. So, terminal point B (-3,2) Example: If u [1,3), D-2,4l, what is 2u 3i? Solution 20 3 20,31-30-2,41 C2,61 -6,127 C2 6), 121 Eg,-61 Vectors in three dimensions The preceding discussion readily extends into three dimen- sions. Definition: We write R3 or the set of all ordered triples of real numbers. Example 1.6: If 2, 11 and U 3, 2, compute v and draw it in the ane. Solution: W- V C2- (-3), 4-2 Cs,-11 (NB Remark: Note that for any points A and B in the ay-plane, we have This is equivalent to the earlier formula OB OA+A AB OA Example 1.7: If the vector 1,3 is drawn in the zy-plane with initial point A (-2,-1), what is its terminal point? Solution: OA AB OB OB AB OA. So, terminal point B (-3,2) Example: If u [1,3), D-2,4l, what is 2u 3i? Solution 20 3 20,31-30-2,41 C2,61 -6,127 C2 6), 121 Eg,-61 Vectors in three dimensions The preceding discussion readily extends into three dimen- sions. Definition: We write R3 or the set of all ordered triples of real numbers.The elements of R3 are again called vectors, and we express them using either row or column notation: a1 a1, a2, a3 Or a2 a3 There is again a visual perspective: We introduce a "right- handed coordinate system" with three mutually perpendicular axes labelled ar, y and 2. The vector la a2, a3 should be nterpreted as representing the displacement from the origin O to the point (a1, a2, a3) in this coordinate system: a, x displacement displacement az at ay z displacement More generally, the displacement from a point A a1, a2, a3) to another point B (b1, b2, b3) determines a vector a2, b3 a3 Arithmetic: The arithmetical concepts of the previous sec- tion extend in the obvious way: Let il 01, 2, ugl, v1, v2, v3 be vectors in R3 and let c e R be a scalar. Do 0,01 (zero vector) ii+ if vi, u2 v2, v13 val (addition). ci cvi, cv2, curs] scalar multiplication -if vi, v2, -val (negation). ii-U lu1 -v1, u2 v2, us v3) subtraction. The visual interpretation is the same as before, and previous identities (e AB BC AC, AB OB -OA) remain valid. Vectors in n dimensions We now generalize to the case of n dimensions, where n is any natural number. Since the case n 24 escapes our visual sense, we
More Less

Related notes for MATH 125

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit