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7/17/2019 Math 54 Reviewer 3rd LE
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University of the Philippines Chemical Engineering Society, Inc. (UP KEM)Math 5 ! "r# $ong E%am &evie'er
Cartesian coordinate system in three dimensionsPoint – represented by an ordered triple of realnumbers (x, y, z)
Coordinate axes – three perpendicular linespassing thru the Origin, O (x, y, and z axes)Octant – division by the three coordinate planes(xz, yz and xy plane) into eight parts (rst octant
shon belo)!idpoint formula – to nd the midpointconnecting the line segment beteen to points
M =( x1+ x2
2,
y1+ y2
2, z1+ z2
2)
"phere – set of all points hose distance frompoint C is r
r2=( x−h )2+ ( y−k )2+( z−l )2
#here r is the radius (h, $, l) is the coordinates of the centre C
%ectors – &uantity that has both magnitude anddirection, geometrically represented by an arro,and denoted by a letter ith an arro above it' he vector from the origin to the point (a, b,
c) is represented by the vector ⟨ a ,b ,c ⟩
' *n v=⟨a , b , c ⟩ a, b and c are the
components of v
' + vector v can be represented by any
directed line segment that has same
magnitude and direction as v
' v=⟨a , b , c ⟩∧u= ⟨d , e, f ⟩ +re e&ual if and
only if a=d ,b=e ,∧c=f
et v be the vector from point P (a, b, c) to
- (d, e, f) hen v=⟨d−a , e−b , f −c ⟩ and
ritten as vector PQ .
' *f third component of vector is zero, e thin$of it as a vector in the xy'plane
.orm of a vector – the magnitude of a vecto
v=⟨a , b , c ⟩
‖v‖=√ a2+b2+c
2
/nit vector – a vector of length or norm 0 1
2ero vector – v=0= ⟨0,0,0 ⟩
3irectional angles – +ngles that a vecto
v=⟨a , b , c ⟩ ma$es ith the positive x, y, and z
axes (α , β ,γ )
cosα = a
‖v‖
cosβ= b
‖v‖
cosγ = c
‖v‖
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University of the Philippines Chemical Engineering Society, Inc. (UP KEM)Math 5 ! "r# $ong E%am &evie'er
%ector operations
A ± B= ⟨ a1
± b1
, a2
±b2
, a3
± b3 ⟩
c A=⟨ c a1
, c a2
, c a3 ⟩
‖c A‖=|c|‖ A‖
' *f c>0 , then c A is in the same direction
as A
' *f c<0 , then c A is in the opposite
direction as A
' *f c=0 , then c A=0
"tandard basic vectors
i=⟨1,0,0 ⟩
j= ⟨0,1,0 ⟩
k =⟨0,0,1 ⟩
.ormalizing a vector – turning a vector into a unitvector
uni vecor ~v= v
‖v‖
3ot product – et A= ⟨ a
1, a
2, a
3 ⟩ and
B=⟨ b1
, b2
, b3 ⟩ ,
A ! B=a1
b1+a
2b2+a
3b3
A !
B=‖
A‖‖
B‖cos"
#here " is the angle beteen A and B ,
" ϵ [0,# ]
' o vectors A∧B are orthogonal or
perpendicular if and only if A ! B=0
Properties4
1) A ! B=B ! A
5) A ! c B=c ( A ! B)
6) A (B ± $ )=( A ! B ) ± ( A ! $ )
7)
A !0=0
8) A ! A=‖ A‖2
Pro9ections ( %ro jab) ' perpendicular
pro9ection of the position representation of
b
onto the line containing the position
representation of a
%ro j a b=a !b
‖a‖2 a
Cross product ' et A= ⟨ a
1, a
2, a
3 ⟩ and
B=⟨ b1
, b2
, b3 ⟩ ,
A & B=de ( i j k
a1
a2
a3
b1
b2
b3
)¿de (a
2 a
3
b2
b3)i−de (a
1 a
3
b1
b3) j+de (a
1 a
2
b1
b2)k
7/17/2019 Math 54 Reviewer 3rd LE
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University of the Philippines Chemical Engineering Society, Inc. (UP KEM)Math 5 ! "r# $ong E%am &evie'er
A & B=‖ A‖‖B‖sin"
' *f A & B=0 , then the vectors are parallel
' :or A=
(
a1
a2
b1 b2
), de A=a
1b
2−a
2b1
' he vector A & B is orthogonal to both
A and B
Properties4
1) A & B=−B& A
5)c B=c ( A & B)
A &¿
6) A & ( B± $ )=( A & B ) ± ( A &$ )
7) A &0=0
+rea of a parallelogram
A=‖a & b‖=‖a‖‖b‖sin"
"calar triple product
A ! ( B &$ )=( A & B ) !$
%olume of a parallelepiped ' =|a ! (b &c )|
Plane (# ) – determined by a point and a
normal vector
a ( x− xo )+b ( y− yo )+c ( z− zo )=0
#here ⟨ a ,b ,c ⟩ is a normal vector to the plane
( xo , yo , zo) is a point on the plane
ine (l) – determined by a point and a parallevector
a) %ector form
r= ⟨ xo , y o , zo ⟩+⟨ a,b ,c ⟩
b) Parametric form
{
x= x0+a
y= yo+b
z= zo+c
c) "ymmetric form
x− xo
a =
y− yo
b =
z− zo
c
#here ⟨ a ,b ,c ⟩ is a parallel vector to the line
( xo , yo , zo) is a point on the line
*ntersections
1) Of to lines – point; e&uate each componenof parametric e&uations of the to lines, thensolve for the to variables t and s, substitutethen nd the point
5) Of to planes – line; solve for the system o
e&uations of the to planes, let x= and
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University of the Philippines Chemical Engineering Society, Inc. (UP KEM)Math 5 ! "r# $ong E%am &evie'er
solve for y and z in terms of it, express line indesired form
6) Of a line and a plane – point; substitute eachcomponent of the parametric e&uation of theline into the e&uation of the plane, solve for t ,then nd the point
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University of the Philippines Chemical Engineering Society, Inc. (UP KEM)Math 5 ! "r# $ong E%am &evie'er
3istances
1) <eteen points4 P1(a1 , b1 , c1) and
P2(a2 , b2 , c2)
d ( P1, P
2 )=√ ( a1−a
2)2+( b
1+b
2 )2+( c
1+c
2 )2
5) <eteen point P( xo , yo , zo) and plane
# :ax+by+cd+ z=0
d ( P , # )=|a xo+b y o+c zo+d|
√ a2+b2+c
2
6) <eteen point P( xo , yo , zo) and line
l :{ x= x
1+a
y= y1+b
z= z1+c
d ( P , l )=‖⟨ xo− x
1, y o− y
1, zo− z
1 ⟩ & ⟨ a,b ,c ⟩‖√ a2+b
2+c2
7) <eteen to lines – use a point on one of the
lines, then use formula d ( P , l)
8) <eteen to planes – use a point on one of
the planes, then use formula d ( P , # )
=) <eteen intersecting lines or planes – zero
Cylinders – surface that consists of all lines thatare parallel to a given line and pass through a
given curve, the graph of an e&uation of tovariables in 6'3
>o to dra41) ?raph the e&uation in to dimensions5) ?enerate the cylinder by moving it along the
direction of the axis of the missing variable
races ' plane curves that are determined bypro9ecting the surface along a given coordinateplane (x'y, x'z or y'z)
>o to dra41) "ubstitute the e&uation of the plane
( xn=k ) to the e&uation of the surface
5) ?raph the resulting e&uation in todimensions
"urface of revolution – surface generated hen aplane curve is revolved about a xed line
?enerating
curve
+xis of
revolution
"urface
e&uation
y=f ( x)
z=f ( x)
x'axis y2+ z
2=[ f ( x ) ]2
x=f ( y)
z=f ( y)
y'axis x2+ z
2= [ f ( y ) ]2
x=f ( z)
y=f ( z)
z'axis ( z )f ¿¿
x2+ y
2=¿
-uadric sections
1) @llipsoidA"pheroid
(2
( +
(2
( +
(2
( =1
5) >yperboloid of one sheet
(2
( +
(2
( −
(2
( =1
6) >yperboloid of to sheets
(2
( −
(2
( −
(2
( =1
7) @lliptical cone
(2
( +
(2
( −
(2
( =0
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University of the Philippines Chemical Engineering Society, Inc. (UP KEM)Math 5 ! "r# $ong E%am &evie'er
8) @lliptic paraboloid
(2
( +
(2
( ±
(2
( =0
=) >yperbolic paraboloid
(2
( −
(2
( ±
(2
( =0
>o to dra41) ?raph the traces of the &uadric in the x'y, y'z
and x'z planes by letting the unanted
variable be e&ual to zero5) *f the trace produces a point, let the unanted
variable e&ual a value that ill produce a
conic section
6) Combine traces on the three'dimensiona
Cartesian coordinate system