Geometry

7.4 Vectors

Vectors

• A vector is a quantity that has both direction and magnitude (size).

• Represented with a arrow drawn between two points.

AB

Initial PointTerminal Point

Vector AB or AB

Component form of a vector

R

S

Horizontal Component

Vertical

Component6

46,4RS ��������������

Horizontal

Component

Vertical

Component

Notation warning

• (2, 3) is a point on the plane.2, 3 is a vector that can be anywhere

on the plane.(2, 3)Any vector with a

horizontal component of 2 and vertical component of 3 is the vector 2, 3.

Write each vector in component form.

3,1 2, 2

4,12, 3

4,0

Translation by Vectors

• From each given point, draw the vector a, b.

• The terminal points is where the translated points are.

Example

Translate JKL using vector -3, 3.

J

K

L

J’

K’

L’Notice: the vectors drawn from each point to its image are parallel.

Vector Addition

• Vector u = a, b • Vector v = c, d • Vector sum:

• u + v = a, b + c, d = a + c, b + d

Example

• Add u = 4, 1 and v = 2, 5• u + v = 4 + 2, 1 + 5 =6, 6• Graphically:

4, 1

2, 5

6, 6

Example

• u = 1, 2, v = 3, -6, w = -6, 1• Find u + v + w and sketch a graph.

• Solution:1, 2 + 3, -6 + -6, 1 =1 + 3 – 6, 2 – 6 + 1 = -2, -3

Magnitude of a vector: is the distance from the initial point to the terminal point. Use distance formula or Pythagorean theorem to find magnitude.

(3, 1)

(8, 8)

The vector in component form is 7,5

The magnitude 6.874

Direction of a vector relative to east.

R

S

Find the Speed of the ship represented by the given vector.

Use a trig ratio to find the direction or the ship relative to east.

6

4

The direction of a vector is determined by the angle it makes with a

horizontal line.

The given vector represents a ship at sea.

The magnitude represents the speed of the ship.

ExampleThe vector represents the velocity of a ship at

sea. Find the ships speed, then find the direction of the ship is traveling relative to west.

Speed is the magnitude of vector (distance)

The direction of the ship is the angle relative to a horizontal line.

6

5