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.