# Precalc 6.3-6.5: Vectors, Dot Products, and Trig Complex Numbers

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Magnitude

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## Tags and Description

### 39 Terms

1

Magnitude

Length of vector (quantity without direction)

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2

Component form starts at the…

origin

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3

Component form of vectors P = (p1, p2) and Q = (q1, q2)

PQ = (q1-p1, q2-p2) = (v1, v2) = v

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4

Magnitude formula

||v|| = √(q1-p1)² + (q2-p2)² = √(v1²+v2²)

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5

Unit vector formula

v / ||v|| = (1 / ||v||)v

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6

Find unit vector in direction of v = (-2, 5) and verify that the result has a magnitude of 1

v / ||v|| = (-2, 5) / √(-2)² + (5)² = (-2/√29, 5/√29)

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7

if ||v|| = 1, v is a…

unit vector

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8

u + v =

(u1 + v1, u2 + v2)

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9

ku =

k(u1, u2) = (ku1, ku2)

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10

||cv|| =

|c| ||v||

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11

(c + d)u =

cu + du

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12

Parallelogram law

u + v is the resultant vector which is the diagonal of the parallelogram with u and v as its adjacent sides

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13

Linear combination of a vector

v₁i + v₂j

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14

Direction angle formula

tanθ = b/a (counterclockwise from positive x-axis)

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15

Trig form of a vector

v = ||v||cosθi + ||v||sinθj

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16

Resultant vector formula

u + v = w (w is resultant)

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17

Speed/velocity/weight is the…

magnitude

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18

A dot product is a…

scalar

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19

||u||² =

u * u

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20

Angle between two vectors formula

cosθ = (u * v) / (||u|| * ||v||)

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21

Find the dot product (6, 2) * (1,3)

6(1) + 2(3) = 12

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22

If and only if 2 vectors A and B are scalar multiples of one another, then they are…

parallel

u = (u1, u2), v = (v1, v2)

ku = v

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23

Alternative form of dot product

u * v = ||u|| ||v|| cosθ

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24

Two vectors u and v are orthogonal (perpendicular) if…

u * v = 0

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25

In force problems, F =

w1 + w2

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26

F =

Gravity + weight of the object directly down from the ramp

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27

w1 =

Force to keep boat from rolling down ramp (arrow going backward on the ramp)

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28

w2 =

Force against ramp (downward perpendicular arrow from the ramp)

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29

Equation setup for finding a plane’s resultant speed and direction

R = P + W

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30

Equation setup for finding what a pilot needs to set the speed and direction to

P = R - W

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31

Absolute value of z = a + bi

|a + bi| = √(a² + b²)

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32

The trig form of the complex number z = a + bi is

z = r(cosθ + isinθ) where

a = rcosθ

b = rsinθ

tanθ = b/a

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33

Modulus

r

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34

Argument

θ of z

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35

Steps to convert complex to trig form

1. Find r through √a² + b²

2. Find θ through tanθ = b/a

3. Put into trig form: r(cosθ + isinθ)

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36

Write the complex number z = 5 - 5i in trig form

r = |5 - 5i| = √5²

tan

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37

DeMoivre’s Theorem

zⁿ = [r(cosθ + isinθ)]ⁿ = r(cosnθ + isinnθ)

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38

Steps to use DeMoivre’s Theorem from complex form

1. Convert to trig form

2. Apply theorem

3. Solve

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39

Find vector v with magnitude ||v|| = 8 and same direction as u = (5, 6)

v = 8(1/||u||)u

||u|| = √25 + 36 = √61

v = 8(1/√61)(5, 6)

v = (40/√61, 48/√61)

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