Vectors in the Plane and in Three-Dimensional Space
Vectors in the Plane and in Three-Dimensional Space
Points in Three-Dimensional Space
y
x
z
(2,3,4)
Distance Between Two Points
Open Ball
000 ,, zyx
of
Closed Ball
000 ,, zyx
of
Directed Line Segment
A
B
Opposite
A
B
Example
Definition of Vector
Example
x
y
Definition of Vector
A vector is a quantity that is determined by a magnitude and a direction.
Speed vs. Velocity
Vector and Scalar Quantities
• Velocity• Displacement• Momentum• Force• Torque• Acceleration
• Speed• Distance• Voltage• Temperature• Time• Volume• Mass
Computer GraphicsRobot Arms
Cartesian Coordinate Represenatation of a Vector
Find the coordinate representation.
x
y
Find the coordinate representation.
x
y
(3,2)
Cartesian Coordinate Represenatation of a Vector
Definition
Vector Addition
y
x
z
(2,3,4)
(1,-2,2)
Vector Addition
y
x
z
(2,3,4)
(1,-2,2)
(3,1,6)
Vector Addition
y
x
z
(2,3,4)
(1,-2,2)
(3,1,6)
Vector Addition
y
x
z
(2,3,4)
(1,-2,2)
(3,1,6)
Vector Subtraction
Vector Subtraction
Vector Subtraction
Vector Subtraction
Zero Vector
0,0,0
Scalar Multiplication
Scalar Multiplication
x
y
z
Length of a Vector
Special Unit Vectors
Special Unit Vectors
x
y
z
i
jk
Unit Vector Representation
y
x
z
Direction of a Vector
Parallel Vectors
The Triangle Inequality
Properties
Application
Problems
• Show that the diagonals of a parallelogram bisect each other.
• Show that the line through the midpoints of adjacent sides of a parallelogram bisect one of the diagonals in the ratio 1:3.
• Show that the quadrilateral whose vertices are the midpoints of the sides of an arbitrary quadrilateral is a parallelogram.