Chapter 6

6.3 – 6.5 Vectors

6.3 Vectors in the PlaneVectors have an Initial Point and a Terminal Point.

They travel in a direction. LABELS: u, v, w or PQ

To have equivalent vectors, you must have the same slope and the same magnitude. Magnitude is the length.

Component Form of a Vectorv – Standard Position Initial Pt. (0, 0) to Terminal Pt. 1 2( , )v v

v = 1 2,v v Magnitude = 2 21 2v v v Magnitude=1,

Unit Vector

Given 2 points, Find Component Form 1 2( , )P p p 1 2( , )Q q q

1 1 2 2,q p q p Magnitude???

6.3 cont’d.Example: Find Component Form and Magnitude of (4, 7) ( 1,5)to

Basic Vector Operations

1 2

1 2

,,

u u uv v v

1 1 2 2

1 2

,

,

u v u v u v

ku ku ku

Given:2,53,4

vw

Find:1. 2v2. w – v3. v + 2w4. 3w – 5v

Finding Unit Vectors:vv

Length = 1, Example 2,5

Standard Unit Vectors 1,00,1

ij

1 2,v v v Is the same as:

1 2v i v j

EXAMPLES…….

6.4 Vectors and Dot ProductsDOT PRODUCT:1 2

1 2

,,

u u uv v v 1 1 2 2u v u v u v

Properties p.422, please look at them!!!!!Examples, find the Dot Product

4,5 2,32, 1 5,3

Given:

1,32, 41, 2

uvw

Find:( )2

3

u v wu vu w

Magnitude = dot u u u

Angle between 2 vectors

cos u vu v

Find angle btwn4,33,5

uv

Vectors are orthogonal if dot product is zero.Parallel????

6.5 Trig. Form of a Complex NumberComplex plane – real axis and imaginary axis a bi

Absolute Value of a complex number is its length. 2 2a bi a b

Example: Plot -2 +5i and find its absolute value.

Trigonometric Form: z a bi is the same as (cos sin )z r i

Where:cossin

a rb r

2 2

tan

r a bba

r is the modulus of z and is the argument of z.

Examples: 2 2 38(cos( ) sin( ))

3 3

z iz i

go to Trig Formgo to Complex Form

Properties p. 434, if you need them I will give them to you.