Assignment 2: Math Review

Assignment #2
CS 184: Foundations of Computer Graphics page 1 of 3
Point Value: 15 points
Fall 2011
Due Date: Sept. 6th, 1:00 pm
Prof. James O’Brien
UNIVERSITY OF CALIFORNIA
The purpose College of this Engineering, assignment UNIVERSITY Department is to test of your Electrical OF prerequisite CALIFORNIA
Engineering math background. and ComputerIn Sciences general, you should
find this College Computer assignment of Engineering, Science to be fairly Division: Department easy. Fall This 2006 ofassignment Electrical CS-184 Foundations Engineering must be done of and solo. Computer To Graphics submit, Sciences please slide under
Prof. O’Brien’s door at 527 Soda Hall.
Computer Science Division: Professor: Fall 2006 CS-184 James O’Brien Foundations of Computer Graphics
Professor: Assignment James O’Brien 1
Introduction: Some of these questions Point may look a little difficult. If they do, consider reviewing how dot
Assignment value: 40 pts 1
products and cross products are formed, how matrices are multiplied with vectors and with each other,
Point value: 40 pts
how determinants are formed and what eigenvalues are. This exercise could take a very long time if
This homework must be done individually. Submission date is Friday, September 8th 2006, at
you don’t think carefully about each problem, but good solutions will use one or two lines per question.
10:00am
Many This homework under the
textbooks must door
supply be of
expressions done 633 individually.
Soda Hall
for the Submission area of a triangle date isand Friday, for the September volume of 8tha 2006, pyramid, at which
you 10:00am Introduction: may under find very theuseful.
door Some of of 633 these Soda questions Hall may look a little di⇥cult. If they do, consider reviewing
Introduction: how dot productsSome and cross of these products questions are formed, may lookhow a little matrices di⇥cult. are multiplied If they do, with consider vectors reviewing
how eachdot other, products how determinants and cross products are formed are formed, and what howeigenvalues matrices areare. multiplied This exercise with vectors could take and
and
with
You a must write neatly. If the assignment cannot be read, it cannot be graded. If you find that you
with veryeach longother, time if how youdeterminants don’t think carefully formed about andeach what problem, eigenvalues but good are. This solutions exercise willcould use about take
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with LaTeX. If you are feeling particularly perverse, consider formatting your solutions using Microsoft’
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now. previous years. To be honest, if you need to cheat on this assignment then just go drop the class
now. Question 1: Two vectors in the plane, i and j, have the following properties (i · i means the
dotQuestion product between 1: Two i and vectors i): i in · i the = 1, plane, i · j = i0, and j · j j, = have 1. the following properties (i · i means the
dot product between i and i): i · i = 1, i · j = 0, j · j = 1.
1. Is there a vector k, that is not equal to i, such that: k · k = 1, k · j = 0 What is it Are
1. there Is there many a vector vectors k, with that these is notproperties
equal to i, such that: k · k = 1, k · j = 0 What is it Are
2.
there
Is there
many
a vector
vectors
k such
with
that:
these
k
properties
· k = 1, k · j = 0, k · i = 0 Why not
3.
2.
If
Is
i
there
and j
a
were
vector
vectors
k such
in
that:
3D, how
k · k
would
= 1, k
the
· j =
answers
0, k · i
to
=
the
0 Why
above
not
questions change
3. If i and j were vectors in 3D, how would the answers to the above questions change
Question 2: For three points on the plane (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) show that the determinant
Question of 2: For three points on the plane (x 1 , y 1 ⇥), (x 2 , y 2 ) and (x 3 , y 3 ) show that the determinant
of
⇧
⌃⇥
⇤ x 1 2 y 1 2 1 ⌅
x 1 y 1 1
⇧
3 3 ⌃
⇤ x 2 y 2 1 ⌅
is proportional to the area of the triangle whose x 3 ycorners 3 1 are the three points. If these points lie on
ais straight proportional line, to what theisarea theof value the triangle of the determinant whose cornersDoes are the thisthree give points. a usefulIftest these to points tell whether lie on
three a straight points line, lie what on a line is theWhy valuedoofyou thethink determinant so Does this give a useful test to tell whether
three Question points lie 3: on The a line equation Why do of you a line think the so plane is ax + by + c = 0. Given two points on the
plane, Question show how3: toThe findequation the values of of a line a, b, inc for the the plane lineis that ax + passes by + c through = 0. Given thosetwo twopoints points. onYou
the
may plane, find show thehow answer to find to question the values 2 useful of a, b, here. c for the line that passes through those two points. You
mayQuestion find the answer 4: Let toequestion 1 = (1, 0, 20), useful e 2 = here. (0, 1, 0), e 3 = (0, 0, 1). Show that if {i, j, k} is {1, 2, 3},
{2, 3, Question 1}, or {3, 1, 4: 2}, Let then e 1 = e i (1, ⇤ e0, j 0), = ee k 2
, where = (0, 1, ⇤0), is ethe 3 = cross (0, 0, product. 1). ShowNow thatshow if {i, that j, k} if is {i, {1, j, 2, k} 3}, is
{2, {1, 3, 1}, 2}, {3, or {3, 2, 1} 1, 2}, or {2, then 1, 3}, e i ⇤then e j = e i e⇤ k , ewhere j = e⇤ k . is the cross product. Now show that if {i, j, k} is
{1, 3, 2}, {3, 2, 1} or {2, 1, 3}, then e i ⇤ e j = e k .

Question 2: For three points on the plane (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) show that the determinant
Question of 2: For three points on the plane (x 1 , y 1 ⇥), (x 2 , y 2 ) and (x 3 , y 3 ) show that the determinant
of
x 1 y 1 1
⇧
⌃⇥
⇤
Assignment #2
x 12 y 12 1 ⌅
CS 184: ⇧
Foundations of Computer Graphics page 2 of 3
3 3 ⌃
⇤ x 2 y 2 1 ⌅
Point Value: 15 points
Fall 2011
is proportional to the area of the triangle whose
x 3 y
corners 3 1
are the three points. If these points lie on
Due Date: Sept. 6th, 1:00 pm
Prof. James O’Brien
ais straight proportional line, to what theisarea theof value the triangle of the determinant whose cornersDoes are the thisthree give points. a usefulIftest these to points tell whether lie on
three a straight points line, lie what on a line is theWhy valuedoofyou thethink determinant so Does this give a useful test to tell whether
three Question points lie 3: on The a line equation Why do of you a line think the so plane is ax + by + c = 0. Given two points on the
plane, Question show how3: toThe findequation the values of of a line a, b, inc for the the plane lineis that ax + passes by + c through = 0. Given thosetwo two points. onYou
the
may plane, find show thehow answer to find to question the values 2 useful of a, b, here. c for the line that passes through those two points. You
mayQuestion find the answer 4: Let toequestion 1 = (1, 0, 20), useful e 2 = here. (0, 1, 0), e 3 = (0, 0, 1). Show that if {i, j, k} is {1, 2, 3},
{2, 3, Question 1}, or {3, 1, 4: 2}, Let thene 1 = e i (1, ⇤ e0, j 0), = e k 2 , where = (0, 1, ⇤0), is ethe 3 = cross (0, 0, product. 1). ShowNow thatshow if {i, that j, k} if is {i, {1, j, 2, k} 3}, is
{2, {1, 3, 1}, 2}, {3, or {3, 2, 1} 1, 2}, or {2, then 1, 3}, e i ⇤then e j = e i e⇤ k , ewhere j = e⇤ k . is the cross product. Now show that if {i, j, k} is
{1, 3, 2}, {3, 2, 1} or {2, 1, 3}, then e i ⇤ e j = e k .
Question 5: For four points in space (x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
the Question determinant5: ofFor four points in space ⇤ (x
x 1 , y
1 x 1 , z
2 x 1 ), (x
3 x 2 , 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
the Question determinant5: ofFor four points in space 4
⇤ (x 1 , y 1 , z 1 ), (x 2 , ⌅y 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
the Question determinant5: ofFor four points in space y y y y ⇤ x(x 1 1 x, y 21 , zx 13 ), (x x 42
, ⌅y 2 , z 2 ), (x 3 , y 3 , z 3 ), (x 4 , y 4 , z 4 ) show that
⇧
the determinant of
z z z z ⌃
y y y y ⇤
x 1 x 2 x 3 x 4 ⌅
⇧
xy z 1 1 1
1 xy z 2 xy z 3 xy z 4
⌃
is proportional to the volume of the tetrahedron
⇧
yz 1 yz 1 2 whose
yz 1 3 yz 1
4 corners
⌃
⌥
are the four points. If these points
lie on a plane, what is the value of the
⇧
determinant
z 1 z1 2 z1 3 z
Does
1 4 ⌃
is proportional to the volume of the tetrahedron this give a useful test to tell whether
four given points lie on a plane Why do you
1
think
1
whose
1
so
1
corners are the four points. If these points
lie is proportional on a plane, what to theisvolume value of the of the tetrahedron determinant whoseDoes corners thisare give thea four useful points. test to If tell these whether points
four lie is proportional Question given a plane, points what 6: tolie the The on isvolume equation the a plane value of of Why ofathe tetrahedron plane dodeterminant you space think whose is so ax Does corners + by this + are cz give the + d afour = useful 0. points. Given test to three If tell these points whether points
four lie space, ongiven ashow plane, points howwhat to liefind on is athe plane value valuesWhy of ofthe a, do b, determinant you c, d think for theso
plane Doesthat thispasses give athrough useful test those to three tell whether points.
four
Question 6: The equation of a plane in space is ax + by + cz + d = 0. Given three points in
You
space, Question may givenfind points
show how
the 6:
lie
to
answer The
on
findequation a plane
the
to
values
question of
Why
of a a, plane 4do useful you
b, c, ind space here. think
for theis so
plane ax + that by + passes cz + d through = 0. Given those three three points points.
You space, Question may show findhow the 7: 6: toanswer Let The findp equation 0
the , to p 1
values question , p of 2 be of athree a, plane 4 useful b, c, distinct ind space here. for the points isplane ax in + that by space. + passes cz Now + d through = consider 0. Given those thethree cross three product points.
n You space, = (p may show 0
find phow 1 ) the ⇥ (p toanswer find 0 the 2 ). toWhat values question does of a, 4 this useful b, c, vector d here. for the mean plane geometrically that passes through Let p be those any three point points. on the
YouQuestion 7: Let p 0 , p 1 , p 2 be three distinct points in space. Now consider the cross product
n
plane
= Question
may formed find by the
(p
p 0 , p 1 and p 2 ; what is the geometric relationship between n and p p 0 (look at
0 p 1 ) ⇥ 7:
answer
(pLet 0
p 0 2
,
to
). p
question
What 1 , p 2 be does three
4 useful
thisdistinct here.
vector points mean geometrically in space. Now consider Let p bethe anycross pointproduct
on the
n the
plane = Question dot (pformed 0 product) p 1 ) by ⇥ 7: (p pLet Show that, if p = (x, y, z) the equation of the plane must be n.(p ) = 0.
0
0 , pp 0 1 and 2 ,). pWhat 1 p, p 2 ; 2 what be does three isthis the distinct vector geometric points meanrelationship geometrically in space. Now between consider Let np and bethe any p cross point p 0 (look product on the at
n
the plane = Question dot
(pformed 0 product)
p 1 ) by ⇥ 8: (p p Suppose Show 0 , p 1 pand 2 that,
). that What p 2 if ; what Ap does
= is(x, ais this square y, the z)
vector geometric the matrix, equation
meanand relationship geometrically
of itsthe inverse plane between and must
Let transpose np beand n.(p
any p exist, point p 0 )(look and =
on
0.
the are at
equal; the plane dot formed
Question
aproduct) matrix by
8:
with p 0 Show , p
Suppose
these 1 and that, properties p 2 if ; what p = (x,
that A is a
is is
square
called y, the z) geometric the an equation
matrix,
orthonormal relationship of the
and its inverse
matrix. plane between
and
Show mustnbe transpose
that, and n.(p for p
exist,
any p 0 )(look angle = 0. at
and are
,
the matrix M( ), defined by the array below, is orthonormal.
equal; Question
dot product)
a matrix8: with Suppose
Show that,
these properties that
if
A
p =
is
(x,
aissquare y, z) the
called an matrix,
equation
orthonormal and
of
its
the
inverse
plane
matrix. and
must
Show transpose
be n.(p
that, forexist, p 0 )
any angle and
= 0.
are,
the equal; Question
matrix a matrix M(
8:
), with defined
Suppose theseby properties that
the
A
array
is ais below,
square called is an matrix,
orthonormal.
and ⇥ its inverse matrix. and Show transpose that, forexist, any angle and are,
the equal; matrix a matrix M( ), with defined theseby properties the arrayisbelow, called cosis anorthonormal.
sin matrix. Show that, for any angle ,
sin cos
⇥
the matrix M( ), defined by the array below, cosis orthonormal. sin ⇥
cos
sin cos
⇥
Show also that M( 1 + 2 ) = M( 1 )M( 2 ); cos use sin this cos sinto argue that the inverse of M( ) is M( ).
Finally, for the point p = (0, 1), what geometrical figure is given by taking the points given by
Show also that M( 1 + 2 ) = M( 1 )M( 2 ); use sin this cos to argue that the inverse of M( ) is M( ).
M( )p for every possible value of
Finally,
Show also
for
that
the
M(
point 1 +
p = 2 )
(0,
= M(
1), what 1 )M(
geometrical 2 ); use this
figure
to argue
is given
that the
by taking
inverse
the
of M(
points
) is
given
M(
by
).
M(
Finally, Show Question also
)p for
forthat every
the9: M( point
possible You 1 + p are = 2 )
value given (0, = M( 1),
of four what 1 )M(
vectors geometrical 2 ); use in the thisplane, figure to argue xis 1
given and thatxthe by 2 , btaking inverse 1 and bthe of 2 , and M( points you ) isgiven M( are told by ).
that M( Finally, )p there foris every the a matrix point possible p M= value such (0, 1), that of what Mxgeometrical 1 = b 1 and Mx figure 2 = isb given 2 . Nowby iftaking you arethe given points a vector given xby
3 ,
Question 9: You are given four vectors in the plane, how M( )p do for youevery determine possible Mxvalue 3 (hint: of show how to find out what 1 and M is 2 from 1 and the data, 2 and you are told
and then apply
that
Question
there is matrix
9: You are
such
given
that
four
Mx
vectors
it to x 3 ).
1 in
1 and
the plane,
Mx 2 x 1 and
2 Now
x 2 ,
if
b 1
you
and
are
b 2
given
, and you vector
are told
3 how
thatQuestion do
there
you
is
determine
a matrix 9: You
Mx
Maresuch given
3 (hint:
that four
show
Mx vectors 1
how
= bin 1
to
and the plane,
find
Mx
out 2
what
= x 1 b 2 and . Now x
is 2 ,
from
if b 1 you and
the
are b
data, 2 given , and you
and
a
then
vector are told
apply
x 3 ,
it
how that
toQuestion do there 3 ).
youisdetermine a matrix 10: Write MxM such down 3 (hint: that the parametric show Mx 1 how = b 1 toequation and findMx out 2 of what = the b 2 . points MNow is from on if you athe line are data, that given and passes athen vector through apply x 3 ,
it how two topoints do x 3 ). youindetermine space, p Mx 1 and 3 p(hint: 2 ; nowshow writehow down to find the parametric out what Mequation is from the of the data, points and then on a apply plane
it Question 10: Write down the parametric equation of the points on line that passes through
that to contains x these two points and a third point, p 3 . Show how to use these parametric equations
two
Question 3 ).
points in space,
10: Write and some inequalities 1 and
down on the parameters 2 the
now
parametric
write down
equation
the parametric
of the points
equation
on a
of
line
the
that
points
passes
on
through
plane
twoQuestion to specify (a) all the line that lie between
that
points
contains these
space, 10: Write
two
p 1 and down
points
p 2
and
; the nowparametric write
third point,
downequation the p 1 and p 2 and (b) all the points on the triangle whose 3 parametric of the points
Show how to
equation a
use these
of line the that
parametric
points passes onthrough
equations
a plane
two thatpoints contains these space, two p points and a third point, p vertices are p 1 , p 2 and p 3 .
and some inequalities 1 and p
on the parameters 2 ; now write down the
to specify 3 . parametric Show how to equation use these of the parametric points on equations a plane
(a) all the points on the line that lie between
that and 1 and Question some contains inequalities these 2 and (b) 11: two
all Mon the ispoints athe points symmetric parameters and a third
on thereal topoint, triangle 2x2 specify matrix. p (a)
whose 3 . Show
vertices It allhas the howtwo points to
areeigenvalues, useon these parametric line 1 2 and ⇥ 0
that
3 and ⇥lieequations
1 . between
pand 1 and some p 2 inequalities and (b) all the on the points parameters on the triangle to specify whose (a) vertices all the points are p 1 , on p 2 the andline p 3 . that lie between
p 1 and Question 11: is symmetric real 2x2 matrix. It has two eigenvalues,
• If pboth eigenvalues are positive, show that, for any 2D vector x, 0 ⌅ x T Mx ⌅ 0 and (max(⇥ 1 Question 2 and (b) all the points on the triangle whose vertices are p
11: M is a symmetric real 2x2 matrix. It has two eigenvalues, 1 , p 2 and p
⇥ 3 . 0 , ⇥ 1 ))x T 0 and ⇥ 1 . x.
Question 11: M is a symmetric real 2x2 matrix. It has two eigenvalues, ⇥ 0 and ⇥ 1 .

the
equal; Question dot product)
a matrix8: with Suppose Show that,
these properties that if Ap = is(x, aissquare y, z) the
called an matrix, equation
orthonormal andof itsthe inverse plane
matrix. and must
Show transpose be n.(p
that, forexist, p
any 0 )
angle and = 0. are,
the equal; Question matrix a matrix M( 8: ), with defined Suppose theseby properties that theAarray is aisbelow, square called is an matrix, orthonormal.
and its inverse matrix. and Show transpose that, forexist, any angle and are,
the equal; matrix a matrix M( ), with defined theseby properties the arrayisbelow, called is anorthonormal.
⇥ matrix. Show that, for any angle ,
the Assignment matrix M( ), defined #2by the arrayCS below, 184: cosis Foundations orthonormal. sin ⇥
of Computer Graphics page 3 of 3
cos
Point Value: 15 points
Fall 2011
sin cos sin
⇥
cos sin cos sin
Show Due Date: also that Sept. M( 6th, 1:00 pm
Prof. James O’Brien
1 + 2 ) = M( 1 )M( 2 ); use sin this cos to argue that the inverse of M( ) is M( ).
Finally, Show also forthat the M( point 1 + p = 2 ) (0, = M( 1), what 1 )M( geometrical 2 ); use thisfigure to argue is given that the by taking inversethe of M( points ) isgiven M( by ).
M( Finally, Show)p also for forthat every the M( point possible 1 + p = 2 value ) (0, = M( 1), of what 1 )M( geometrical 2 ); use thisfigure to argue is given that the by taking inversethe of M( points ) isgiven M( by ).
M( Finally, )p for forevery the point possible p = value (0, 1), of what geometrical figure is given by taking the points given by
Question 9: You are given four vectors in the plane, x 1 and x 2 , b 1 and b 2 , and you are told
M(
thatQuestion )p for every
there is a matrix 9: possible You
M
arevalue such
givenof that
four
Mx
vectors 1 = b
in 1 and
the plane,
Mx 2 =
x 1 b
and 2 . Now
x 2 ,
if
b 1 you
and
are
b 2 given
, and you
a vector
are told
x 3 ,
how
thatQuestion do
there
you
is
determine
a matrix 9: YouMx Maresuch 3 given (hint:
that four show
Mx vectors 1 how
= bin 1 to
and the findplane, Mx
out 2 what
= x 1
b 2 and .
M
Now
is x 2 from , if b 1
you and the
are
data, b 2
given , and andyou a
then
vector areapply
told x 3 ,
it
how that to
do there x 3 ).
youisdetermine a matrix Mx M such 3 (hint: that show Mx 1
how = b 1
toand findMx out 2
what = b 2 . MNow is from if you theare data, given anda then vector apply x 3 ,
it how to
Question
do x 3 ). you determine
10: Write
Mx
down 3 (hint:
the parametric
show how to
equation
find out
of
what
the points
M is from
on a
the
line
data,
that
and
passes
then
through
apply
it
two
toQuestion points
x 3 ).
in space, 10: Write
p 1 and
down
p 2 ;
the
now
parametric
write down
equation
the parametric
of the points
equation
on a
of
line
the
that
points
passes
on
through
a plane
two
thatQuestion points
contains
in
these
space, 10: two Write p 1 and
points down p 2 and
; the now
aparametric write
third point,
downequation the
p 3 .
parametric
Show of the howpoints to
equation
use these a of line the
parametric that points passes on
equations through a plane
two that
and points some
contains
inequalities these space, two pon points
the parameters
and a third
to
point,
specify
p 3 .
(a)
Show
all the
how
points
to use these
the
parametric
line that lie
equations
1 and p 2 ; now write down the parametric equation of the points onbetween
a plane
that p
and 1 and
some contains p
inequalities 2 and these (b) all twothe onpoints the parameters and onathe third triangle
topoint, specify
whose p (a)
vertices
all the points
are p 1 ,
on
p
the 2 and
line
p 3 .
that lie between
3 . Show how to use these parametric equations
pand 1 and p 2 and (b) all the on the triangle whose vertices are p 1 , p 2 and p 3 .
Question
some inequalities
11: M
on
is a
the
symmetric
parameters
real
to
2x2
specify
matrix.
(a)
It
all
has
the
two
points
eigenvalues,
on the line
⇥
that
0 and ⇥
lie
1 .
between
p 1 and Question p 2 and (b) 11: all Mthe is apoints symmetric on thereal triangle 2x2 matrix. whose vertices It has two areeigenvalues, p 1 , p 2 and p⇥ 03 . and ⇥ 1 .
• Question If both eigenvalues 11: M isare a symmetric positive, show realthat, 2x2 matrix. for any 2D It has vector twox, eigenvalues, 0 ⌅ x T Mx ⇥⌅ (max(⇥ 0 , ⇥ 1 ))x T x.
• If both eigenvalues are positive, show that, for any 2D vector x, 0 ⌅ x T 0 and ⇥ 1 .
Mx ⌅ (max(⇥ 0 , ⇥ 1 ))x T x.
If one eigenvalue is zero, show that there is some vector ⇧= such that Mx 0.
• If both one eigenvalue eigenvaluesisare zero, positive, show that showthere that, is forsome any 2D vector vector x ⇧= x, 0such ⌅ x T that Mx Mx ⌅ (max(⇥ = 0. 0 , ⇥ 1 ))x T x.
If one eigenvalue is positive and one is negative, show that there is some vector ⇧= such
• If one
that eigenvalue T is
Mx is zero, positive showand that onethere is negative, is some show vectorthat x ⇧= there 0 such is that someMx vector = 0. x ⇧= 0 such
• If
that
one
x T eigenvalue
Mx = 0
is positive and one is negative, show that matrix is positive definite if, for any vector x, T there is some vector x ⇧= 0 such
Mx 0. Assume that is
•
symmetric,
Athat matrix x T Mx M= real,
is 0 positive
and positive
definite
definite.
if,
What
for any
can
vector
you say
x,
about
x T Mx
its
>
eigenvalues
0. Assume that M is
•
symmetric,
A matrix M
real,
is positive
and positive
definite
definite.
if,
What
for any
can
vector
you say
x,
about
x T Mx
its
>
eigenvalues
0. Assume that M is
symmetric, real, and positive definite. What can you say about its eigenvalues
Question 12: M is a 2x2 matrix.
• Show that the eigenvalues of M are the roots of the equation 2 trace(M) + det(M) = 0.
Something Optional: +5 pts
If you decide to continue into graduate school, you will most likely have to write papers about
your graduate research work. If these papers contain any math, you will need to set your equations
using a tool like L A TEX, Mathematica, or MS Word’s Equation Editor. (Listed in order of my own
preference.)
If you use one of these tools (or something similar) to write up the solutions to at least 4
questions, you will get two extra credit points on this assignment. If you do the entire assignment
with the tool you will get a total of five points extra.
My personal preference is for L A TEXbecause it produces the nicest looking output. L A TEX’s
downside is that it is hard to learn, although once learnt it is fast to use. Mathematica and Word