Vectors & Trigonometric Form

By: Ina Salvaleon, Hannah Galt, Irem Guney

Topics

Properties of vectorsOperations of vectorsChanging to standard formApplications of vectorsChanging to trigonometric form

Formulas– Vectors

Component form: Magnitude and direction:

Angle between two vectors: Unit vector of v: Dot product: Linear combination form: Projection vector: Work:

,x y ( cos ), ( sin )mag mag 2 2mag x y

arctan( )yxdir

0,

180 ( )

if x

add

|| || (|| ||)cos u v

u v

|| ||

v

v

1 2 1 2u v x x y y x yi

2 ( )vu v

proj u vv

work F PQ ��������������

cos

PQ

F PQ

proj F PQ

��������������

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Vectors

u= < 5, -4 > v= < -3, 6 >

Find exact magnitude:

Find direction angle of v: arctan (6/-3) + 180° = 116.57°

2 25 4u 41

Vectors

u= < 5, -4 > v= < -3, 6 >

Vector parallel to u:

Vector orthogonal to v:

<-3, 6> switch and change one sign

<-6, -3>

5, 4 2 10, 8

Vectors

u= < 5, -4 > v= < -3, 6 >

Find a unit vector in the same direction as u:

< 5, -4 >/ = < 5 / , -4 / > Find a unit vector in the same direction as u,

but with a magnitude of 7:

< 5 / , -4 / > (7) =

<35 / , -28 / >

41 41 41 41 41

41 41 41 41

41 41 41 41

Vectors

u= < 5, -4 > v= < -3, 6 >

Find the dot product of u and v:

x1x2+y1y2= (5)(-3)+(-4)(6) = -39

Find the angle between u and v:

cos 39 /( 41 3 5) 155.22

Vectors

u= < 5, -4 > v= < -3, 6 >

Find the projection of u onto v:

(-39/45) <-3,6> = <117/45, -234/45>=

<13/5, -26/5> Component of u that is orthogonal to projvu:

w2 = u-w1 = <5, -4> - <13/5, -26/5> =

<12/5, 6/5>

Vectors

Change 6-3i to trig form (radians):

Change to standard form:

2 26 3 3 5r

3 5 ( .46)cis

arctan( 3 / 6) .46

5(cos(5 6) sin(5 6))i

5cos(5 / 6) 5sin (5 / 6) ( 5 3 / 2) (5 / 2)i i

Application of Vectors

A wagon full of children with a combined weight of 110 pounds is on a hill with an incline of 9 degrees. What force will someone have to exert to keep it

from rolling down the hill? What is the force perpendicular to the hill?

A: 17.2 pounds B:108.6 pounds

Application of Vectors

Three vectors with magnitudes of 150, 200, and 100 act on an object at angles of 40°,-80°, and 120°, respectively with the positive –axis. What is the resultant vector? What is its magnitude and direction?

A: 99.64, 13.94 B: 7.96 352.04

Application of Vectors

A tractor pulls a log 800 meters and the tension in the cable connecting the tractor and the log is approximately 14,000 newtons. What is the work done if the direction of the force

is parallel to the horizontal? What is the work done if the direction of the force

is 20° above the horizontal?

A: 11,200,000W Nm B: 10,524,557.35W Nm

Formulas– Trigonometric Form

Trig form: Standard form: Multiplying: Dividing: Exponents: Roots:

1 2 1 2 ( )r cis r cis r r cis 1

1 2 2 ( )rr cis r cis cisr

( ) ( )n nrcis r cis n 2( )knnn

rcis r cis 0,1, 2...( 1)k n

rcisx yi

Multiplying in Trig Form

3 3 5 57(cos sin ) 8(cos sin )

4 4 3 3i i

3 57 8

4 3cis cis

1. Simplify to

2. Multiply 7 times 8

3. Add and 3

4

5

3

2956

12cis

Solution:

Dividing in Trig Form

7 712(cos sin )

6 6

3(cos sin )3 3

i

i

712

6

33

cis

cis

1. Simplify to

2. Divide 12 by 3

3. Subtract by76

3

54

6cis

Solution:

Powers of Complex Numbers

82 i

1. Find the radius and angle.

2. Plug in the radius and angle into the powers formula.

2 2(2) (1) 5 r 12arctan 0.46

8 8

5 0.46 5 8 0.46cis cis

Powers of Complex Numbers

Solution in trig (radians) and standard forms:

625 ( 3.68) 536.58 320.48cis i

82 i

Roots of Complex Numbers

Find the 5th roots of 2 3( )i

1. Find the radius and angle.

2. Plug in the radius and angle into the roots formula.

2 22 3( ) ( ) 7r 3

2arctan 0.71 2.43

105 2.43 2

57( 2.43) 7 kcis cis

Roots of Complex Numbers

Find the 5th roots of 2 3( )i

3. Plug in the values of 0, 1, 2, 3, & 4 for k. Make sure your calculator is in radians!

0

1

2

3

4

k

k

k

k

k

10 2.43 2 0

5

10 2.43 2 1

5

10 2.43 2 2

5

10 2.43 2 3

5

10 2.43 2 4

5

7

7

7

7

7

cis

cis

cis

cis

cis

Roots of Complex Numbers

Find the 5th roots of 2 3( )i

Solution in trig (radians) and standard forms:

0

1

2

3

4

k

k

k

k

k

10

10

10

10

10

7 0.49 1.07 0.57

7 1.74 0.21 1.20

7 3.00 1.20 0.17

7 4.26 0.54 1.10

7 5.51 0.87 0.85

cis i

cis i

cis i

cis i

cis i

Homework

Ch. 6 Review (pg. 480) 6.3: #43, 59, 71 6.4: #77, 85, 97(just part a, projection of u onto v) 6.5: #107, 115, 119