Vectors

Lesson 13.4Pre-AP Geometry

Lesson Focus

This lesson defines the concept of a vector. Vectors have important applications in physics, engineering, and other applied sciences.

Basic Terms

VectorA directed quantity that has both magnitude and direction.

MagnitudeA scalar value having physical units. The magnitude of a mathematical object is its size or absolute value.

DirectionThe spatial relation between something and the course along which it points or moves.

Vector

Vector

Vector AB shows the distance and direction from point A to point B.

The magnitude of vector AB is the distance from A to B. The magnitude of a vector is like the absolute value of a number. It is never a negative number.

The direction to B from A is found by determining the x-component and the y-component of the vector.

Vector

• Two vectors are equal if they have the same magnitude and the same direction.

• Two vectors are parallel if they have the same or opposite directions.

• Two vectors are perpendicular if they have perpendicular directions.

Multiplying by a Scalar

• Multiplying a vector by a scalar multiplies the magnitude of the vector.

• If the scalar multiple is negative, then the direction of the vector is reversed.

Vector Sum

Geometer’s Sketchpad Examples

01 Vector Basics02 Vector Basics03 Vector Basics04 Vector Addition07 Vector Length (Magnitude)

Example #1

Given: P(-3, 4) and Q(-2, -2)Sketch vector PQ, then find the magnitude and direction of vector PQ.

Solution:

6,142,32 PQ

3736161 22 PQ

Magnitude:

Direction:

Example #2

Given: P(-3, 4) and Q(-2, -2)Find: and

Solution:

PQ3 PQ2

18,363,136,133 PQ

12,262,126,122 PQ

Example #3

Show that (6, -3) and (-4, 2) are parallel.Solution:

The slope of (6, -3) is -1/2. The slope of (-4, 2) is -1/2.

The slopes are equal so the vectors are parallel.

Example #4

Show that (6, -3) and (2, 4) are perpendicular.Solution:

The slope of (6, -3) is -1/2.The slope of (2, 4) is 2.

Since -1/2 2 = -1, the vectors are perpendicular.

Is a vector a ray?

No

If B is between A and C, then ray AB and ray AC are the same ray, but vector AB vector AC.The magnitude of vector AC is greater than the magnitude of vector AB.

Written Exercises

Problem Set 13.4A, p.541: # 2 - 26 (even)