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8-1-1958
Some applications of vectors to the study of solid
geometry
Ossie Malinda Smith
Atlanta University
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SOME APPLICATIONS OP VECTORS TO THE STUDY OP
SOLID GEOMETRY
A THESIS
SUBMITTED TO THE FACULTY OP ATLANTA UNIVERSITY
IN PARTIAL PULFILLMENT OP THE REQUIREMENTS FOR
THE DEGREE OP MASTER OP SCIENCE
BY
OSSIE MALINDA SMITH
DEPARTMENT OP MATHEMATICS
ATLANTA,
GEORGIA
AUGUST 1958
if
1
-r
ACKNOWLEDGMENTS
* ""
36
The writer acknowledges her indebtedness to those
instructors who made the writing of this thesis possible.
Among those who should be mentioned are the late Dr.
Gokhale, Mrs.
G. C. Smith,
Dr.
V. Bohun,
and Dr. L.
V.
Cross.
I am grateful to each for whatever contribution was made.
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
±1
LIST OP FIGURES
iv
Chapter
I. INTRODUCTION
!
II. FUNDAMENTAL DEFINITIONS
Definition of a Vector.
*
Vector Symbolism
Negative and Null Vector....
Equality of Vectors
Geometric Representation of a Vector...
Reciprocal Vectors
3
3
3
3
4
4
4
III. ALGEBRAIC OPERATIONS
Vector Addition,
Subtraction of Vectors
Scalar Multiplication
Collinear and Coplanar Vectors
Decomposition of Vectors
7
7
8
IV. VECTORIAL OPERATIONS
Product of 5Bwo Vectors
Triple Products
Vector Product of Pour Vectors
9
9
10
11
V. GEOMETRIC APPLICATIONS
Applications to the Sphere
Applications to the Tetrahedron
VI.
CONCLUSIONS
5
5
6
12
12
13
17
VII. BIBLIOGRAPHY
18
iii
LIST OP FIGURES
Page
1. A Vector.
5
2. Negative Vectors
3
3. Reciprocal Vectors.
4
4. Vector Addition.
5
5. Demonstration of the Commutative Law...
5
6. Demonstration of the Associative Law
6
7. Subtraction of Vectors
6
8. Cross Product of Two Vectors
10
9. A Sphere with Center, C, and an Arbitrary
Point, P, on the Circumference
11
10. Diagram Used to Prove That the Joins of
Opposite Edges of a Tetrahedron Bisect
Each Other.
14
.11. Diagram Used to Show That the Relation
PA' - PB« - PC' - PD» - 1 Holds if P Is a
TP
Mr
CU'
EE
Point within Tetrahedron ABCD and A», B', C,
and D' Are the Meets of the Lines Joining the
Vertices to P and the Opposite Paces
iv
16
CHAPTER I
INTRODUCTION
We differentiate between two types of mathematical quan
tities;
those having magnitude only are called scalars; those
having both magnitude and direction are called vectors.
In
this paper some of the elementary properties of the latter and
their application to three dimensional space will be presented.
Vectors arose as a result of the mathematicians1 attempts
to represent complex numbers geometrically,
A first step in
this direction was the invention of the quaternion in 1843 by
Sir William Hamilton,
He made use of four quantities in rep
resenting the quaternion:
units, i, 3, and k.
a + bi + e5 + dk.
the units:
one scalar and the three imaginary
The quaternion, q is written in the form
The following laws govern multiplication of
i2-J2:k2: -1; U = -3i r. k; fr = -*$ : H
ki r -ik - j.
Quaternions are easily adapted to matrix algebra.
matrix
(a 4 bi
c 4 di\
-c + di
a - bi/
)
The
may also be written in the following
i
<V
If the coefficients of b, c, and d are called i, j, and k
respectively, we obtain the form a+bi+oj+dk.
The products
2.
of the units in this expression behave ;just as those of the
laternions of Hamilton.
lUaternions
f
As an example,
we see that ij r
!Ehe use of quaternions was not too widespread,
for the
applied mathematician found them too general and complex,
Josiah Willard Gibbs,
an American,
used some of the ideas
introduced by Hamilton and rejected others in developing the
field known as Vector Analysis.
This paper is an attempt to show the properties of vectors,
apply these properties to problems in three dimensional space,
and draw some conclusions from this study.
CHAPTER II
FUNDAMENTAL DEFINITIONS
Definition of a vector.- A vector is defined as a quan
tity having both magnitude and direction.
Associated with
every vector ar© two points- the initial point or head and
the terminal point or tail.
Graphically, we show the vector
PQ as the directed line segment from P to Q,
■p
Figure 1
Vector symbolism.- The vector above may be indicated by
the symbol P§ in which the magnitude is the scalar quantity
PQ , and the arrow shows the direction to be from P to Q.
In the proofs of theorems, however, this method becomes
unwieldly; thus, hereafter a vector will be denoted by a
single small letter with a bar to distinguish it from a
scalar quantity.
Negative and null vectors.- The set of vectors which
have the same magnitude but opposite direction of any given
vector is called the negative of the given vector.
6L
Figure 2
3.
Thus, in
4.
figure 2, s«
= -a*.
In the case Mien the initial point and
the terminal point coincide, then we have what is called the
zero or null vector,
The magnitude of this set of vectors
equals zero*
Equality of vectors.- Equal vectors are defined to "be
those having the same magnitude and the same direction.
This
means that all equal vectors are parallel*
Geometric representation of a vector,- We represent a
vector on a Cartesion diagram "by plotting the coordinates of
its initial point and those of its terminal point.
If the
coordinates of the initial point and terminal point of a
are ( x , y , z ) and (x . y , z ) respectively, then -me
differences xg - \$ y2 - yx» a*1* \ " \ are oai:Led the
components of a.
Two vectors are equal if and only if their
corresponding components are equal.
Reciprocal vectors,- If a' is parallel to a but a' • - ,
_____*___-__—_———•»—■—
a,
then a*1 and S are said to be reciprocal vectors.
figure 3
CHAPTER III
ALGEBRAIC OPERATIONS
Vector addition,- Tha algebraic operations of addition,
subtraction,
and multiplication may be performed on vectors.
We define the sum of a and b" in the following manner;
the vectors, without changing their directions,
place
so that the
terminal point of a coincides with the initial point of b~;
the sum is the vector whose initial point is the head of a
and whose terminal point is the tail of F.
1
Hgure 4
By using the definition we can show that the commutative
law for vector addition holds, i.e., a + b" s b" + a.
of a and b" is c\
The sum
By completing the parallelogram in figure 5,
we get b" - a - c also; therefore, jfche law holds.
6.
Ihe associative law for vector addition states that for
any vectors, a, F, and c", a f (F 4 c) s (a 4. F) + "5.
6, (F ♦ c) = cT and a ♦ 3 - 5.
c s e,
Thus,
In figure
Likewise (a ♦ F) s T and T 4
the associative law holds.
t KLgure 6.
The sum of several vectors is obtained by using the
definition repeatedly.
This sum is the vector whose head is
the initial point of the first addend and whose tail is the
terminal point of the last addend.
Subtraction of vectors.- Subtraction of vectors may be
considered as the special case in which we add a vector to
the negative of another vector.
Then a - b = a + (-b).
Gibbs £l;12j gives a neat geometric interpretation of the
subtraction of two vectors.
are parallelograms.
and a f (-F) » <T.
In figure 7» OAB'D and OABC
By the definition of addition, a+b s c
However, <T equals and is parallel to e;
therefore "a - b~« e".
/SL
7.
The rule states that the difference between two vectors drawn
from the same origin is a vector whose initial point is the
tail of the vector representing the subtrahend and whose
terminal point is the tail of the vector representing the
minuend.
Scalar multiplication.- The product of a scalar quantity,
s, by a is a vector which has been increased s-times in magni
tude.
If s is positive,
the same as that of a;
opposite that of a.
s s 0.
the direction of the new vector is
if s is negative,
the direction is
The product sa yields the zero vector if
When a vector is multiplied by a scalar,
each of its
components is multiplied by that scalar.
Both the right and left distributive laws hold for the
multiplication of a vector by a scalar,i.e.,
a(s
+ s )
12
(s
4 s )a s
- s a + s a.
12
Collinear and coplanar vectors.- If there exist scalar
quantities, s and t(not both zero), such that sa + tF - 0,
then a and F are called collinear vectors.
In order for three vectors to lie in the same plane they
must be linearly dependent.
This says scalar quantities, r,
s, and t (not all zero), must exist such that ra + sE + tc =0.
Pour vectors are always linearly dependent.
8.
Decomposition of vectors.- Coffin jT2;21 J shows that
any vector may be decomposed into any number of component
vectors which may or may not lie in the same plane.
A most
convenient selection is three mutually perpendicular axes, x,
y, and z, along which are placed the unit vectors I", J, and E".
A vector,
a, with components a
,
a
.
a
12
3
as a r -fa J 4 a £, a quite useful form.
12
3
,
can then be expressed
The vector is then
said to be a linear combination of the unit vectors.
CHAPTER IV
VECTORIAL OPERATIONS
Product of two vectors.- In multiplying one vector by
another, one encounters two types of products, scalar and
vector products.
These products have no analogue in the
algebra of real numbers.
The scalar product, sometimes called
the inner product, is indicated by a raised dot; the vector
product, also known as the cross product is indicated by an x.
We define the inner product of two vectors as the product
of their lengths multiplied by the cosine of the angle between
them.
By using the definition we get the following rules for
multiplication of the units:
3E * EJ * O; W - k¥ - 0.
I
=7 = E
- lj iq sji s 0;
If two vectors, a and F, are expres.
sed as linear combinations of their components and the unit
vectors, we get the scalar product in the following way:
(a.F) - (a T ♦ a 7 +a Ic).(b t + b 7 + t) E).
1
2
3
1
2
3
combining terms on the right gives (a b
may conclude
Multiplying and
* a b
•» a b ).
We
that the scalar product of two vectors is the
sum of the products of corresponding components.
The cross product of a and F is the vector whose magni
tude is ab sin 9 (where 9 is the angle between a and F) and
whose direction is the perpendicular from the plane of a
and F viewed counterclockwise(Figure 8).
9.
10.
Multiplication of the units is given by the following
rule:
I
* J
-I5E - 'J,
= £ -<£? 3PJ - -$T = E;
/Note that I
^ =
r X x T etc.)
All of the laws governing multiplication of real numbers
do not hold for the latter product,
29ais product is not com
mutative, but the distributive and the scalar associative laws
are valid.
(ax*)
Figure 8,
iEriple products,- It is possible to multiply three vectors
and obtain either the triple scalar or the triple vector pro
duct*
lor any three vectors, a", "E>,
and c,
expressions of the
type a(Fxe") and (axF)c are known as triple scalar products
of a, F, and c",
Shis product can be written in terms of the
components of the three vectors.
pression, we get (a.r
(e_r 4 ©pi * cJs:)J.
By expanding the first ex
+ a«7 f aj£)-j Cb,r 4 bgj 4 b-lc) x
Simplifying the expression in the
bracket we obtain (a I 4 a]u fc)./(bc -be )T +
1
2
3L23
32
(bo -be )] Mb c - b c )Tsl
31
13
12
21
Ihis last form reduces to
a(be-bc)4a(bc-be)*a(bc-be),
123
3 2231
13
312
21
Asa deter-
minant the scalar product takes the following forms
11.
b
a
a
3
b
3
c
3
for any three vectors a, F, and c, an expression of the
form ax (F x c) is called a triple vector product.
This
product could also be expanded in terms of its components
to give the form
(a b
+ a b
(a c
4& c
4a © )(kfci +^0 + b^k) -
1 1. 2 2. 3 3- l
+ a b )(c i * o j4 c,k)»
5
%
Mor® ooncisely, this
is written in the form(a»gb -(a*tje.
Vector product of four vectors.- She importance of the
triple scalar product lies in the fact that all higher pro
ducts can be reduced to the triple product.
vector product (axb)x(cxd).
Consider the
By making use of a
reduction formula we may write the following two expressions:
c(abd)- d(abo) and b(acd) - a(bcd).
two
When combined, these
expressions give a(bcd) - b(acd)+c(abd) - d(abc) - 0,
whioh is the condition for four vectors to be linearly de
pendent/, 5;12 J .
CHAPTER V
GEOMETRIC APPLICATION
A few problems involving well-known properties of the
sphere and tetrahedron have been solved as a means of showing
how vectors are applied to Solid Geometry.
Problem I;
Express the equation of a sphere in vectorial
form.- A sphere is completely determined if its radius and
"Hie position of its center are known.
In the sphere,
S,
(ligure 9) suppose P is a point on the sphere and C is the
center.
Let the coordinates of P be (p:., p , p
1,, 2
C be (c , c , c ).
12
(p - c )
22
(p
3
2
2
- c ).
3
We may then writeISFf- iaj= 7 (p
3
- (p
3
) and those of
ll
,
-c )
3
2-re
&
/ , or/al-
J
'
(p
2
L
- c ) i (p
11
1
-c )
2
1
2
- c ) +
22
This is the equation of the sphere in terms of
vectors.
ilgure 9
12.
-
13.
Problem II:
Show that the
joins of the midpoints of
opposite edges of a tetrahedron bisect each other.-
Let A, B,
0, and D be the vertices of a tetrahedron, and let a, b", c", and
<I be the respective position-vectors of the vertices with
respect to an arbitrary origin, 0,
Let X, m, n, u, v, and w
represent the position-vectors of the midpoints of the edges
with respect to 0 (Figure 10).
Adding vectors a and b" gives
the sum AB since the diagonals of a parallelogram are equal.
However, AB ■- 2n; therefore n-_a__£_F.
2
obtain these additional equations:
In a similar way, we
.
I - b" + c, m - c" + at
u - a » <I , v - b" 4 cT , and w - c" f cC.
!Qie limes which we
want to bisect each other are LU, HW, and MY.
The midpoints
of these lines are the points of intersection of the diagonals
of parallelograms OLDX , ONWX , and OMVX respectively, where
12
3
X , X , and X represent the unknown vertices of the completed
1
2
3
parallelograms.
It is now necessary to show that these mid
points are coincident.
are
Ihe position-vectors of these points
Iu , 1+v, and n ■» w, each of which is equal to the
expression •a»T)fc»dT
Therefore, the lines bisect each other.
14.
V
T>
\
O
ELgure 10.
15.
Problem III.
Show that the expression
PA»
AA*
Jjj7 s 1
PB«
PC«
+ —— +
BB»
00'
holds if P is a point within tetrahedron ABCD and
A1, B1, C, and D' are the points where the lines joining P
to each vertex of the tetrahedron intersect the opposite face,The four vectors involved may be reduced to three by choosing
an arbitrary vertex, say A, as the origin, and then writing
b" s AB, c s AC, and 3 = AD (Figure 11). AP5p can be written
as a linear combination of b", c, and <T; then p - rF 4sc 4 tcf,
where r, s, and t are not all zero.
Since AA»
- a1
is a scalar
multiple of p, a' = k(rb" * so 4> tcT); also a1 - F 4 BA>, Since
BA* is coplanar with (c - b) and (d - b), it may be expressed
in terms of them.
Then a1 = m(c - b) 4 n(d - "b) 4 b.
equating the coefficients of b,
c, and d from the two expres
sions of a', we obtain k - 1 - m - n
r
which when combined give k <-
»
k - m
~ s •
1 •»
that PA1 s. AA1
- AP.
PA1 _ k - 1 -l-(r + s + t).
AA« "
k
we find that PB_ _ r
BB»
- (k - lJrfT + so 4 t?).
Using the same method
PCJ. . s, ^d PD_ . t.
00*
- %
Substituting the values of AA1
and AP and simplifying, we find that PA1
Then
and k - n
Erom the figure we
"FTTTT *
see
By
DD*
four values gives the required results.
Addition of these
16
fi
CHAPTER VI
CONCLUSIONS
Since quantities such as force,
velocity,
acceleration,
etc* have direction and magnitude associated with them,
the
applied mathematician will find vectors to be a most desir
able aid.
The pure mathematician is able to revel in the
fact that the conversion of theorems from conventional form
to vectorial form is possible; but this conversion, in many
cases,is not a simplification.
Both the usefulness and beauty of vectors, when considered
together,
serve to make them an integral part of the field
of mathematics.
17
BIBLIOGRAPHY
1
Josiah Willard aibbs, Vector Analysis, lew Haven, Conn.,
Yale University Press, 1913
2
Joseph George Coffin, Vector Analysis, lew York, John
3
Albert Potter Wills, Vector and Censor Analysis, Hew
Wiley & Sons, 19051
York,
Prentice-Han,
inc.,
19)1
4
Homer E. Newell, Vector Analysis, Hew York, McGraw-Hill
5
Dr. Lonnie Cross,
Book Company, liiu.,
19 55.—
" Hotes for Introduction to Higher
Geometry", 1957.
18
(Mimeographed)