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

Magnitude

Length of vector (quantity without direction)

Component form starts at the…

origin

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

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

Magnitude formula

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

Unit vector formula

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

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)

if ||v|| = 1, v is a…

unit vector

u + v =

(u1 + v1, u2 + v2)

ku =

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

||cv|| =

|c| ||v||

(c + d)u =

cu + du

Parallelogram law

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

Linear combination of a vector

v₁i + v₂j

Direction angle formula

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

Trig form of a vector

v = ||v||cosθi + ||v||sinθj

Resultant vector formula

u + v = w (w is resultant)

Speed/velocity/weight is the…

magnitude

A dot product is a…

scalar

||u||² =

u \* u

Angle between two vectors formula

cosθ = (u \* v) / (||u|| \* ||v||)

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

6(1) + 2(3) = 12

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

Alternative form of dot product

u \* v = ||u|| ||v|| cosθ

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

u \* v = 0

In force problems, F =

w1 + w2

F =

Gravity + weight of the object directly down from the ramp

w1 =

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

w2 =

Force against ramp (downward perpendicular arrow from the ramp)

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

R = P + W

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

P = R - W

Absolute value of z = a + bi

|a + bi| = √(a² + b²)

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

z = r(cosθ + isinθ) where

a = rcosθ

b = rsinθ

tanθ = b/a

Modulus

r

Argument

θ of z

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)

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

r = |5 - 5i| = √5²

tan

DeMoivre’s Theorem

zⁿ = \[r(cosθ + isinθ)\]ⁿ = r**ⁿ**(cos***n***θ + isin***n***θ)

Steps to use DeMoivre’s Theorem from complex form

1. Convert to trig form
2. Apply theorem
3. Solve

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)