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Calc III > Exam 1 > Flashcards

Exam 1 Flashcards

1
Q

where are octants situated (R2 or R3)

A

R3

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2
Q

how to find the angle between vectors

A

cos-1 (u*v)/(|u||v|)

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3
Q

what are skewed lines

A

not parallel and they do not intersect

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4
Q

positive orientation

A

gotten by letting t increase from a to b
defualt orientation of line if not specified
put arrow going counter-clockwise

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5
Q

what vector is perpendicular to all other vectors

A

0 vector

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6
Q

when is torque at max

A

when sin =1 so angle is pie/2

when angle is pie/2, r and F are perpendicular

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7
Q

what does |u|cos equal

A

(u*v)/ |v|

dot product of u and v divided by magnitude of v

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8
Q

what is the absolute value of a vector mean

A

the length of U (magnitude of U) (norm of U)

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9
Q

normal vector

A

a vector that is perpendicular to the plane

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10
Q

what is parallel to a vector

A

the vector multipled by a scallar
V || KV

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11
Q

what does it mean when K is greater than/equal to 0

A

vector whose direction is that of u and whose magnitude is K(U) (magnitude of u)

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12
Q

what to do when UxV are not parallel

A

draw parallelogram, find area of parallelogram (drop right angle); get |v||u|sin

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13
Q

when does the point hit the line

A

when the distance from point Q to line L is less than the two radians combined

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14
Q

how to find the distance between points

A

draw a right triangle, draw another one, use pythag theorem

the first line down is (z-c)
the next line down (second triangle) is (x-a)
the horizonta line (connecting 2nd trinagle to og point) is (y-b)

The points are (a,b,c) and (x,y,z)
line down in first triangle goes to (x,y,c)
the next line down leads to (a,y,c)

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15
Q

properties of cross products

A

UxV= -(VxU)
|UxV|= |v|u|sin
(au) x (bv)= (ab)(UxV)
Ux(v+w)= (UxW)+(VxW)

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16
Q

Cross product

A

UxV

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17
Q

how are projvU and scalvU related

A

|projvU| = |scalvU|

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18
Q

how to find the direction of a vector

A

u/|u| = (1/|u|)(u)
opposite direction is making it negative

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19
Q

when is line parallel to vector

A

P.P ||v
P.P=tv for some scalar t

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20
Q

sphere

A

center (a,b,c) and radius r
(a,b,c) any real number, r is greater than 0

(x-a)^2 +…….= r^2

completing the square

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21
Q

how to move from R2 to R3

A

<x,y>
R3 is <x,y,z>
so R3 is <x,y,0>

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22
Q

YZ plane

A

x=0
goes down, right, sky without stopping

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23
Q

how to find points where origin intersects the coordinate axes

A

find the intercepts for x,y,z

set the other two to 0 in equation and solve)

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24
Q

(v/|v|)

A

<cos, sin>
if not on origin <a+cos, b+sin>

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25
Q

coordinate notation

A

<x,y>

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26
Q

what is scalvU

A

scalar component of u in dir of v
|u|cos

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27
Q

projvU

A

orthogonal direction of U onto V
how much of u points in direction of v

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28
Q

how many answers are there to what is perpendicular to u and v

A

infinitely many because scalars are involved
k(UxV)

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29
Q

how to find area of triangle

A

make a parallelogram

triangle OQP

.5(area of parallelogram)= .5 (|OP x OQ|)

combine OP and OQ

then do matrix for the cross product

then multiply by .5

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30
Q

Identities with scalars and vectors

A

U,V,W are vectors; K,L are scalars

U+V=V+U
(U+V)+W=U+(V+W)= U+V+W
U+0=U=0+U
U+-U=0=U-U
K(LV)=(KL)U
K(U+V)=KU+KV
(K+L)U=KU+LU
1U=U
0U=0
K0=0

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31
Q

what is torque

A

when force F is applied at tip of r

creates a TWISTING motion at origin

angle is N*m

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32
Q

when giving equation for line what is t

A

0<t<1

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33
Q

what is area of parallelogram

A

|v||u|sin
the area of parallelogram is wqual to the magnitude of the cross product |Uxv|

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34
Q

How are vectors represented

A

an arrow
has direction (way its pointing)
the length is magnitude of vector

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35
Q

how to find a line through a point (P.= x.,y.,z.) that is perpendiclar to the vector <a,b,c,>

A

P(x,y,z) any point on l

OP= OP. +OP

4 different equations one can use:
<x,y,z>= <x.,y.,z.> + tv
r= r. +tv
<x,y,z>= <x.,y.,z.> + t(a,b,c)
<x,y,z>= (x.t+at, y.t+bt, z.+ct)

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36
Q

what happens when different planes have the same normal vector

A

the planes are parallel

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37
Q

parallel planes

A

n1 and n2 are parallel
n1||n2

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38
Q

what does U+V mean, U-V, KU

A

<u1+v1; u2+v2; u3+v3>
<u1-v1; u2-v2; u3-v3>
<Ku1, Ku2, Ku3>

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39
Q

how many equations are there for one line

A

infinitely many

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40
Q

when is torque the largest

A

when r and F are perpendicular

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41
Q

example of vectors

A

acceleration, force, velocity

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42
Q

how to find the distance of a point to a line

A

|PQxV|= |v|d

d= (|PQxV|)/|V|

angle is (d)/(PQ)= sin

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43
Q

what is a degenerate case

A

when k=0
the 0 scalar is parallel to any vector
(the direction is also the same)

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44
Q

what happens when one vector is 0 in a cross product

A

the cross product will equal 0

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45
Q

what coordinate vectors are perpendicular to each other

A

any two are perpendicular to each other
(2 of i,j,k)

46
Q

plane def

A

infinite and extend in all 4 directions

47
Q
A
48
Q

how to find vectors (AB)

A

<b1-a1, …..>

49
Q

angle for the resultant

A

tan^-1(y/x)

50
Q

|ku|

A

parallel, same direction as u
k|u|=c
k= (c|u|)(u)
= (|c/|u||)(|u|)=c

51
Q

standard unit vectors

A

i,j,k
standard base vecotrs
coordinate vectors

52
Q

how to find magnitude and direction of cross product

A

magnitude is |u||v|sin
direction is going of something

53
Q

for skewed lines, how to tell lines are parallel or intersect

A

parallel
L1 and L2 are parallel if V1 and V2 are parallel
V1||KV2 for some scalar K
when the ratio for k is the same for x,y,z then it is parallel

intersect
set the two equations equal for each x,y,z
then solve for x,y,z
(set two equation equal, then put in to find the others)

if the last equations do not end up equalling each other, then not intersect

54
Q

how many octants are there

A

8

55
Q

how to find the work of an object

A

|F||d|cos

unit is joule

56
Q

what is unit vector

A

a vector that length=1

57
Q

what happens to orthogonal projections when angle is pie/2

A

projvU=0

58
Q

U*v

A

|u||v|cos
angle between 0 and pie

59
Q

how to know if U and V are parallel

A

the angle is exactly 0 or pie

60
Q

what happens when UxV is zero

A

u and v are parallel

(angle is either 0 or pie)

61
Q

when are two planes orthogonal

A

n1 and n2 orthogonal
n1*n2=0

62
Q

cross products of standard base units

A

(ixj)= k (jxi)=-k
(jxk)= i (kxj)= -i
(kxi)= j (ixk)=-j

63
Q

what happens when angle is between 0 and pie/2 for the orthogonal projections (projvU)

A

|projvU|= |u|cos (v/|v|)
magnitude= (c)(v/|v|)

64
Q

when are two vectors equal

A

(U,V) are U=V if and only if their direction are equal and magnitude equal

CAN have different locations

65
Q

what is |A|

A

detminant of A
det(A)

66
Q

<a,b> * <-b,a> equals what

A

0- orthogonal

67
Q

what is a scalar multiple of U by K

A

KU

68
Q

what determines planes

A

any 3 points that aren’t all on the same line (two different lines going on) determine the plane (that contains them). a point in a plane and a normal vector (n not equal 0) uniquely determine that plane

69
Q

how to find equation when noncollinear

A

PQ x PR
(perpendiclar to the plane and normal vector)

three points P,Q,R

70
Q

how to know when vectors are orthogonal with cross products

A

the cross product of UxV will always be perpendicular to u and v
VxU also perpendicular (-UxV)

71
Q

zero vector in coornidate unit vector

A

<0,0,0> = 0i+0j+0k=0

72
Q

what happens when F1 and F2 are applied at a point to an object

A

their combined effect on object is F1+F2
(the resultant)

73
Q

torque

A

|r||F|sin = |torque|=torque

= |rxF|

74
Q

when are two vectors perpendicular

A

when the angle is pie/2 (90)

75
Q

what does a<t<b mean for curve of r(t)

A

curve has initial vector r(a) and terminal vector r(b) and natural orientation of positive orientation

76
Q

how to figure out cross products

A

do the matrixes

77
Q

what does UxV mean

A

|u||v|sin

78
Q

what does the dot product have to be for the vectors to be perpendicular

A

dot product must be 0

79
Q

how to describe set of points at which Q intersects the yz plane, xy, plane, and xz plane

A

for each plane, one is 0. Set that to 0 for each plane

then remaining create the equation (R2) that you graph

each line is part of the line on the triangle

80
Q

projection of a line of xy plane (z=0)

A

(x,y,0) is a point

81
Q

zero vector

A

vector whose initial point is equal to termina point
length=0
any direction depending on what is needed
(a point)

82
Q

XZ plane

A

y=0
never go right or left

83
Q

what does it mean when k is less than 0

A

vectors whose direction is oppposite of U and whose mangutide is (K)(U)- magnitude of both K and U

84
Q

scalar

A

real number (k)

85
Q

what is R2 version octants

A

quadrant

86
Q

parameter equations for line l

A

x= x.+at
y=y.+bt
z=z.+ct

x.,y.z. is the point
a,b,c is the vector

87
Q

what does r(t) equal

A

r(t)= <x(t), y(t), z(t)>

often thought of as depicting the motion of a point

88
Q

what is d in regards to work

A

the motion of object under force

if two points, its the vector of them combined

89
Q

vectors

A

the movement of a particle along a line segments between 2 points (R2 and R3)

completely determines by direction and magnitude

90
Q

Pythag explanation in finding the distance bw two points

A

(a,b,c) and (x,y,z,)

do (x-a), (y-b), (z-c)
square each paranthesis then add up
then root square the sum

91
Q

how to do matrixes

A

a b
c d

do ad-bc

for i,j,k the numbers are those in the other two lines (if i, its j and k)

+i -j +k

92
Q

what happens with orthogonal projections when the angle is between pie/2 and pie

A

projvU= |u|cos(v/|v|)

93
Q

How to find midpoints

A

average of each point

94
Q

what does projvU equal

A

|u|cos (v/|v|)

((uv)/ (vv)) (v)

95
Q

equations for planes

A

<a,b,c> * <x-x., y-y., z-z.>=0
a(x-x.)+b(y-y.)+c(z-z.)=0
ax+by+cz=d
d= ax.+by.+cz.

point (x.,y.,z.)
normal vector <a,b,c>

96
Q

3 coordinate planes

A

YZ (x=0)
XZ (y=0)
XY (z=0)

97
Q

what does O mean

A

origin

98
Q

identities with dot products

A

uv=vu
u(v+w)= uv + uw
k(uv)= (ku)v= u(kv)
vv= |v|^2
ov=0

99
Q

How are x,y,z situated in regard to each other

A

all are perpendicular to each other
(x and z)
(x and y)
(y and z)

100
Q

1 octant

A

region infinite-naturally divided by x,y,z axis
x,y,z greater than 0

101
Q

graph of r(t)

A

trace at the points and get the curve
(tips of position vecotr- given by values of position function)

102
Q

what does smaller diagonal mean

A

(u-v)
instead of (u+v)

103
Q

the direction of a cross product

A

if zero, the direction is 0

if not zero then the right hand rule

104
Q

what does x(t) equal

A

r(t)= <acost, asint>
= x(t)^2+ y(t)^2= a^2

105
Q

MAKE SURE TO STILL LOOK IN NOTEBOOK- CANT DRAW THINGS

A

also redo the HW and do practice problems in textbook

106
Q

initial and terminal point

A

initial point- tail
terminal point- head (tip)
vector name is in bold)

107
Q

Dot products

A

u=<u1,u2>
v=<v1,v2>

U*V= U1V1+ U2V2

108
Q

find force

A

F=ma
|F|=m|a|
unit is Newton

109
Q

how to find diagonal

A

combine U and V
make parrelogram
diagonal is U+V

110
Q

when are two vectors parallel in regards to cross product

A

it equals zero

111
Q

XY plane

A

z=0
(goes up and slanted in)

112
Q

noncollinear

A

not all together on a line

Calc III (2 decks)

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