12.5 Equations of Lines and Planes
Introduction to 3-space (from the first day of class)
1-space: graph x = 4
2-space: graph x = 4
3-sapce: graph x = 4
Lines
What determines a line?
2-space:
3-space:
The equation of a line
2-space
3-space
Vector Equation
12.3 The Dot Product
Dot Product
If a a1 , a2 , a3 , b b1 , b2 , b3 are vectors, then the dot product of a and b is
Example 1: Find the dot product of the given vectors
A) v 5i 8 j, w i 2 j
B) a 7 j, b 2i 3 j
C)
2, 4 , 3, 1
D) 1, 7, 4 , 6, 2,
1
2
E) v i
12.2 Vectors
Vector
Indicates a quantity having
Represented as a
Denoted in one of three ways: v, v , x, y
Example: v a1 , a2 from A to B
Equivalent vectors v = u or v u
Zero vector 0 or 0
Position vector OP a1 , a2
Example 1: Give the vector
A) From (2
12.1 Three-Dimensional Coordinate Systems
Coordinate Planes
xy-plane
yz-plane
xz-plane
Octants
first octant
Ordered triple (a, b, c)
Projection of point P onto plane
Q(a, b, 0)
R(0, b, c)
S(a, 0, c)
1
Example 1: Plot the points (2, 3, 5) and