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How do you calculate vectors?

octors MATH 2415 Math Lab Review

ų dot product : Scalar 4 length / magnitude : VX1 + X2+X3 y distance blu 2 vectors:

Parameter 1-= V(x,-y,)*+(x2-y.)* +(xz-ys)?

How do you calculate projection?

(tip to tip) - Angle blu 2 vectors: os os

Ž. ū = 1xül cose

Žiūgų = . - Projection

Coeff./scalar of a broj 627 - (g

a

projjx

<1,2,32 x ů = (0)(2) +(2)(1) + (3)(-2) us (21-27

How do you identify a quadric surface?

We also discuss several other topics like Which is the best definition of activation energy?

< <3 3,

golü 3) (distributive) = ñ. ū+ X2 = -2 + We also discuss several other topics like What is the content of cantwell vs connecticut?

3 (1-2) + (2-1) + (3+2)2 Vi+ i + 25 = 1727 We also discuss several other topics like What is the meaning of moral hazard?

i lx

ex. Angie blw 9 + 2 . 7. 2 = 6

= lūll El coso le=(3) (V14) cose ce = cose > We also discuss several other topics like What is the difficulty in producing or comprehending speech not produced by deafness or a simple motor deficit?

0 = cosil

माप

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projzx

= XZ

Ž

=

10<3,2,17

la xb1 = 1a1151 sino

(also area of parallelogram) If you want to learn more check out Who should religious leaders be?

complete

scalar projection)

Vix Te = liikl = 1(2-0)-(43)

12 Y

31 tk(O+1)

1-1 0 21 = <2, -7, > So, l= (2, 1, 1) + t <2, 7, 17

on

line

Planes y

Lines / Planes Para: l(t) = Ē+ t , dincetin vector. If you want to learn more check out Who is paul of tarsus?

to line ex para of line through (1,2,3),

Il to line l,lt) = (1,0,1) + tv

lit) = (1,2,3) + ex: line through (1,9, 2) +(2,-3, 1)

Level

Set

upp pppppppppppppp9999999999987

1) Level Set/Stadard:

Ax + By + C z = d. (A,B,C) ñ (normal vector)

exi

para.

plane contains 3 pts P = (1,2,1) Q = COD) R=(3, 0, 2)

n = PO PR : = <-1-1,07 €2,-2, 13

ex

<2-1, 3-0, 1-27.

<1 -3,-1) = ✓ 21+) = (1, 0,2) - +(!, -3,-1) Line through (2, 1, !) Vi

Given: line at to both 2,4)= (1, 1) + +(2,1,3).

l (t) = (1,0,0) + + (-1,0, 2)

11 Tol <-1, 1, 43 = n, 12 -2 1 l = <a, b, c

CA

1

-x

+

y

+ 4z =

d

sapnas

point

Q

-10 + (1) + 461) = 0 = 5 . 71-x + y + 4z = 51

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: Plane contains line e

l= (2, 1, 3)2 + +(!, , -2) + point: 10, 1, 2) in

P cannot be > on line l

Jax J = ñ

- Z hyperbolic paraboloid

(Saddle ) Cylindncal Coords

(rez) rcoso rsin

Z = z !

2

.

2

-

x

x

VE xv, to!

+ y2 = r2 o = tan

y

for

ŽEZ!

VEL,1,-27 V2 =< - 2, 0,-17

P=1 (+)

Some t 0 - 2 + t

(x)

ex

ñ

=

1 =

1

1+

ĭ - 2

t

(

zer (in cyl.) Sketch Surface. E szer: Vx+y? (cone) L as 7 increases, so

does the radius of the circle.

L

M Quadric Surfaces

s Review equations /

forms

for each .

7

cone

NOT

che

sheet)

Darabad

5

axis determined bu negative

yo trace: 73 7x2

→ Z = 1x x + y = z2 = full cone it and - ex: 7 = r2

→ Z= x2 + y2 (paraboloid)

---

>

coefficient.

40 > x=2

parabola So NOT

= -1

- Z2

C2

profite, cone!

I

hyperboloid of

2 sheets.

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s os 0 st

Curves

Spherical Courds

po, 0) X = Psin coso 4 = P sind Sin@ +- Dcos o

ex: F(t) = (3 cos (tt), 2 sin (et), t ) Sketch. Egn of tangent line at

- (0, 2, 1/2) x= 3 cos(tt) → *13 = cos(Tt) 4= 2 sin(tot) 912 = Sin (it)

anos nas

ex: P = 2 cos ♡

Sketch.

p2= 2 pcose

elliptical cylinder the curve lies the cylinders

on

/ Plot

of

shape.

points to get idea (0) = (3,0,0) (12) = (0, 2)

**+ y2 + z = 27 (sphere!) x2 + y2 + z?- 27+1=0 +1 = x +42 + (7-1)=1 Sphere, center: (0,0,1), radius !

I ex: 05 774 Sketch

egn. of tangeurt line

l(t) = p + to va direction vector) = f'(to) =

5-31tsin(tet), 2 recos (tt td, O. 30oslit) 2 = 2sin ( t) 112 = t

r('12) = -370, 0, !>

for ...

tangent line T: (0,2,112) + (-311,0,1+

anelons

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Arc length

r(t) = (cosct), Sinct), 2+2) length of curve blw points Loo) and coi, 211 312 ) ?

paz | r'(t) I dt = length

Iretol = |(-sinct>" + (cesitz" + (3#") %

a

312.

To