VECTORS IN A PLANE

Pre-Calculus Section 6.3

CA content standards: Trigonometry

• 12.0 Students use trigonometry to determine unknown sides or angles in right triangles.

• 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems.

• 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides.

• 19.0 Students are adept at using trigonometry in a variety of applications and word problems.

OBJECTIVES

• Represent vectors as directed line segments• Write vectors in component form • Add and subtract vectors and represent them

graphically• Perform basic operations on vectors using

scalars • Write vectors as linear combinations of i and j• Find the direction angle of a vector • Apply vectors to real-world problems

Vector

• Directed line segments

• Named by initial point and terminal point (like a ray, in geometry)

Ex: PQ

Q

P

• Vectors have direction and magnitude

• Magnitude = length

• Given the endpoints of a vector use the distance formula to find its magnitude

22qpqp yyxxPQ

• Vectors with the same direction and magnitude are equal.

• Vectors can also be named using a single, bold, lowercase letter

Ex: u=PQ

Given P=(0,0) Q=(3,4) R=(4,3) S=(1,2) T=(-2,-2)a=PQ, b=RP, c=ST, d=QP

which vectors are equivalent?

-2 -1 1 2 3 4

-2

-1

1

2

3

4

x

y

-2 -1 1 2 3 4

-2

-1

1

2

3

4

x

y

-2 -1 1 2 3 4

-2

-1

1

2

3

4

x

y

-2 -1 1 2 3 4

-2

-1

1

2

3

4

x

y

dc

ba

component form

• Standard position – initial point at origin• Component form – use terminal point to

refer to vector

yx ,v vv

component vertical

component horizontal

yv

vx x

y

v

vy

vx

• Zero vector, 0 = <0,0>

• Unit vector 1v

Component form: general position

• Remember equal vectors are determined by direction and magnitude – not location

• Rewrite in standard position

yy pq,pqv xx PQ

x

y

x

y

vector operations: scalar multiplication

• Scalar – number• To multiply a vector by a scalar – multiply

each component by that scalarex 4,3t

t find t5 find t5 find

5

43 22 25

2015 22 20,15

45,35

Vector operations: addition

• To add vectors, add their components

lrlr find 3,6 5,4Ex:

yyxx baba ,ba

8,2

35,64

vector operations: addition

• Visually, vectors can be added using the parallelogram law

– Join vectors tail to head– Resultant vector is diagonal of parallelogram

ex 3,4 25, cm

m3 2- c

Visually and algebraically find

cm cm 2

6,15

23,53

6,8

32,42

1,9

32,45

7,6

322,452

Unit vectors

• Remember a unit vector is any vector with magnitude of 1

• To find a unit vector in the direction of a vector v, divide the vector by its magnitude

Ex. Find the unit vector in the direction of <5,-2>

v

vu

29

292,

29

295

Unit vectors

• A vector can be written in terms of a directional unit vector and its magnitude

uvv

Write in terms of the unit vector w/ the same direction

247v ,

25

24,

25

725 v

standard unit vectors

0,1i

1,0j

Horizontal unit vector

Vertical unit vector

x

y

j

i

• ALL vectors in a plane can be written as a combination of i and j

jiz yxyx zzzz ,

• Ex. W has an intial point at (6,6) and terminal point (-8,3) write it as a combination of i and j

W in component form is <-14, -3>

As a combination of i and j, W = -14i – 3j

direction angles and vectors

• Direction angle is from the positive x axis. • Use right triangle trig.

x

y

vvy

vx

θ

θvx cosv

sinvyv

x

y

v

vtan

Write each vector in component form

30sin7,30cos7

x

y

x

y

78300°

2

7,

2

37

300sin8,300cos8

34,4

30°

x

y

Write the magnitude and direction angle for each vector

ji 94

255,5

315,135

1tan

<-5,5>

9794 ji

962.293

,962.1134

9tan