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Unformatted text preview: Math 173
Kouba
Using Vectors to Determine Equations for Lines (in R2 and R3) and Planes (in R3) EQUATIONS OF LINES IN R2 and R3 I.) Find an equation in parametric form of the line L passing through the point (31:0, yo) and in the direction of (parallel to) vector (a . Let (x, y) be a random point on line L. b
Form vectors (3:0) and . Then
yo 3/ 2 (:0) + t (vector equation of line L) for some t E R —~+
0 .7: ‘_ m0 at
(y) — (yo) + (M) —’
r _ 330 + at
y ‘— yo + bt
. :c 2 x0 + at
L. ' y : yo + bt for t E R (equation of line in parametric form) 11.) Find an equation in parametric form of the line L passing through the point a
(230,340, 2:0) and in the direction of (parallel to) vector b . Let (x,y,z) be a random
0
£170 .17
point on line L. Form vectors yo and y . Then
Z0 Z
a: 330 a
y 2 yo + t b (vector equation of line L) for some t E R —>
Z Z0 C
x :30 at
y = 210 + bt ——>
z zo c
a: 3:0 + at
21 = 1/0 + bt —>
z zo + ct aczm0+at
LI y=y0+bt z z 20 + ct for t E R (equation of line in parametric form) EQUATIONS OF PLANES IN R3 Find an equation of the plane passing through the point (3:0,y0,zo) and which is a
perpendicular to (normal to) the vector N = b . Let (any, 2) be a random point on
c
(I) — $0
the plane. Form vector V 2 y — yo , which lies in the plane. Then
Z — ZO NLV —> N~V=0 ———+ a SITm0
b ' y*’yo =0 ——>
C z—zo a(a:— 270) +b(y—y0) +c(z ~ 20) = 0 ...
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This note was uploaded on 02/17/2012 for the course MATH 17B taught by Professor Kouba during the Winter '07 term at UC Davis.
 Winter '07
 Kouba

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