April 16, 2010
8.6
Vectors in Space
k<0,0,1>
v = <a, b , c >
zero vector: 0 = <0,0,0>
j <0,1,0>
standard unit vectors:
i = <1,0,0>
j = <0,1,0>
k = <0,0,1>
Standard Unit Vector Notation:
v=vi+vj+vk
i<1,0,0>
April 16, 2010
component form:
Q(x,y,z )
v = <x-a, y-b, z-c >
P(a,b,c )
v = <v1, v2, v3>
magnitude:
unit vector:
v
= v + v +v
v
v
dot product: u v = u v + u v + u v
Other Properties (Pg. 690)
April 16, 2010
1. Find the component form and magnitude of vector
v having initial point (3,4,2) and terminal point (3,6,4).
2. Find a unit vector in the direction of v.
April 16, 2010
3. Find the dot product of <1,3,-2> and <4,-2,3>.
April 16, 2010
Just after takeoff a plane is pointed due east. Its air
o
velocity vector makes and angle of 30 with flat ground with
an airspeed of 250mph. If the wind is out of the southeast
at 32mph, calculate a vector that represents the plane's
velocity relative the point of takeoff.
(Let i point east, j point north, and k point up.)
April 16, 2010
How do we get LINES in space?
Two ways:
1. use one vector equation
2. a set of three parametric equations
April 16, 2010
Lines and Planes in Space
In a plane, slope is used to determine the equation of a line.
In space, vectors are used to determine the equation of a line.
P (x, y, z) L
L II v
Po(xo, yo, zo)
v = <a,b,c>
L consists of all pts. P(x,y,z) for which PoP is II to v.
PoP = tv
let r = <x, y, z>
and
r = <x , y , z >
o
o o o
then
PoP = r - ro
r - r = tv
o
r = ro + tv
Vector Form
We can determine Parametric Equations of a Line in Space:
April 16, 2010
Write the vector and parametric forms of the line
through P in the direction of v.
o
Po = (-3,8,-1), v = <-3, 5, 2>
April 16, 2010
Using the standard unit vectors i, j, and k, write a vector
equation for the line containing the points A(2,1,-3) and B
(4,5,-2), and compare it to the parametric equations for the
line.