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 ofof

Closed Ball ofof

Directed Line Segment A B

Opposite A B

Example

Definition of Vector

Example

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 Graphics Robot Arms

Cartesian Coordinate Represenatation of a Vector

Find the coordinate representation.

(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

Zero Vector

Scalar Multiplication

x y z

Length of a Vector

Special Unit Vectors

x y z i j k

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.