Vectors in the Plane and in ThreeDimensional Space

Vectors in the Plane and in Three-Dimensional Space

Points in Three-Dimensional Space z (2, 3, 4) y x

Distance Between Two Points

Open Ball of

Closed Ball of

Directed Line Segment B A

Opposite B A

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.

Find the coordinate representation. (3, 2)

Cartesian Coordinate Represenatation of a Vector

Definition

Vector Addition z (2, 3, 4) (1, -2, 2) y x

Vector Addition (3, 1, 6) z (2, 3, 4) (1, -2, 2) y x

Vector Addition (3, 1, 6) z (2, 3, 4) (1, -2, 2) y x

Vector Addition (3, 1, 6) z (2, 3, 4) (1, -2, 2) y x

Vector Subtraction

Vector Subtraction

Vector Subtraction

Vector Subtraction

Zero Vector

Scalar Multiplication

Scalar Multiplication z y x

Length of a Vector

Special Unit Vectors

Special Unit Vectors z k j y i x

Unit Vector Representation z y x

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.