Vectors in Two and Three Dimensions. Definition A vector is a quantity that is determined by a magnitude and a direction.

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Presentation transcript:

Vectors in Two and Three Dimensions

Definition 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

Arrow Representation

Cartesian Coordinate Represenatation of a Vector

Find the coordinate representation.

(3,2)

Unit Vectors x y z i j k

Unit Vector Representation y x z

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)

The Zero Vector and Additive Inverse

Properties of Vector Addition

Scalar Multiplication x y z

Properties of Scalar Multiplication

Length of a Vector

Example

Definition of Unit Vector

Example

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.