MODERN GEOMETRY · 2018. 9. 19. · MODERN GEOMETRY Plane Coordinate Geometry Ederlina...
Transcript of MODERN GEOMETRY · 2018. 9. 19. · MODERN GEOMETRY Plane Coordinate Geometry Ederlina...
Plane Coordinate Geometry
MODERN GEOMETRYPlane Coordinate Geometry
Ederlina Ganatuin-Nocon
Term 1, AY2013-2014
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Coordinate Plane
Each ordered pair (p1, p2) of real numbers determines exactlyone point P of the plane.
The point (0, 0) is called the origin.
The ordered pair (p1, p2) is the coordinate vector of P .
A vector may be thought of as a line segment directed from
one point to another. The vector (p1, p2) may be viewed as a
line segment from the origin to point P .
We use the terms “vector” and “point” interchangeably.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Vector Space R2
If x = (x1, x2), y = (y1, y2) and c ∈ R, then
x + y = (x1 + y1, x2 + y2) and cx = (cx1, cx2).
These operations are called vector addition and scalarmultiplication.
The vector 0 = (0, 0) is called the zero vector and−x = (−x1,−x2) is the negative of x = (x1, x2).
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Vector Space R2
Theorem
For all vectors x, y , and z and real numbers c and d,
i. (x + y) + z = x + (y + z)
ii. x + y = y + x
iii. x + 0 = x
iv. x + (−x) = 0
v. 1x = x
vi. c(x + y) = cx + cy
vii. (c + d)x = cx + dx
viii. c(dx) = (cd)x
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Inner Product Space R2
Given two vectors x = (x1, x2) and y = (y1, y2), we define
�x , y� = x1y1 + x2y2.
The number �x , y� is called the inner product (or dot product orscalar product) of x and y .
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Inner Product Space R2
Theorem
i. �x , y + z� = �x , y�+ �x , z� for all x , y , z ∈ R2
ii. �x , cy� = c�x , y� for all x , y ∈ R2 and c ∈ Riii. �x , y� = �y , x� for all x , y ∈ R2
iv. If �x , y� = 0 for all x ∈ R2, then y must be the zero vector.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Inner Product Space R2
Remark
a) R2 is a vector space.
b) �·� is bilinear, symmetric, and nondegenerate.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Inner Product Space R2
Norm of a Vector
For any vector x ∈ R2 we define the length or norm of x to be
|x | =�
x21+ x2
2.
Note that
|x |2 = �x , x�
so that norm and inner product are truly related.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Inner Product Space R2
Theorem
The norm function has the following properties:
i. |x | ≥ 0 for all x ∈ R2
ii. If |x | = 0, then x = 0.
iii. |cx | = |c ||x | for all x ∈ R2 and all c ∈ R
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Inner Product Space R2
Cauchy-Schwarz Inequality
Theorem
Cauchy-Schwarz Inequality.For any two vectors x and y in R2 we have
|�x , y�| ≤ |x ||y |.
Equality holds if and only if x and y are proportional (that is,x = ty for some t ∈ R).
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Inner Product Space R2
Corollary to Cauchy-Schwarz Inequality
Corollary
For x , y ∈ R2,|x + y | ≤ |x |+ |y |.
Equality holds if and only if x and y are proportional with anonnegative proportionality factor (that is, x = ty for some t ≥ 0).
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Euclidean Plane E2
If P and Q are points, we define the distance between P and Q by
the equation
d(P ,Q) = |Q − P |.
The symbol E2 is used to denote R2 equipped with the distance
function d .
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Euclidean Plane E2
Theorem
Let P, Q and R be points of E2. Then
i. d(P ,Q) ≥ 0
ii. d(P ,Q) = 0 if and only if P = Q.
iii. d(P ,Q) = d(Q,P)
iv. d(P ,Q) + d(Q,R) ≥ d(P ,R) (Triangle Inequality)
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Lines
A line in analytic geometry is characterized by the property
that the vectors joining pairs of points are proportional.
We define a direction to be the set of all vectors proportional
to a given nonzero vector.
For a given vector v let
[v ] = {tv | t ∈ R}.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Lines
If P is any point and v is a nonzero vector, then
� = {X | X − P ∈ [v ]} (1)
is called the line through P with direction [v ]. We also write (1) as
� = P + [v ]. (2)
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Lines
When � = P + [v ] is a line, we say that v is a direction vector of�.If � is a line and X is a point, the following means to say X ∈ �,
a) � contains X
b) X lies on �
c) � passes through X
d) X and � are incident
e) X is incident with �
f) � is incident with X
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Lines
Theorem
Let P and Q be distinct points of E2. Then there is a unique linecontaining P and Q, which we denote by PQ.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Lines
A typical point X on the line � = PQ is written
α(t) = P + t(Q − P) = (1− t)P + tQ.
This equation may be regarded as a parametric representation of
the line. As t ranges through the real numbers, α(t) ranges overthe line. The parameter is related to distance along � by the
formula
d(α(t1),α(t2)) = |t2 − t1||Q − P |.
If X = (1− t)P + tQ, where 0 < t < 1, we say that X is between
P and Q.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Lines
Theorem
Let P, X , and Q be distinct points of E2. Then X is between Pand Q if and only if
d(P ,X ) + d(X ,Q) = d(P ,Q).
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
LinesMidpoint of a Segment
Let P and Q be distinct points. The set consisting of P , Q and all
points between them is called a segment and is denoted by PQ.
P and Q are called the end points of the segment. All other
points are called interior points.If M is a point satisfying
d(P ,M) = d(M,Q) =1
2d(P ,Q),
then M is a midpoint of PQ. It can be shown that
M =1
2(P + Q).
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Lines
If � and m pass through P , we say that they intersect at P and
that P is their point of intersection.
Theorem
Two distinct lines have at most one point of intersection.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Orthonormal Pairs
Two vectors v and w are said to be orthogonal if �v ,w� = 0.
If v = (v1, v2), we define v⊥ = (v2,−v1). Clearly, v and v⊥ are
orthogonal and have the same length. Also
v⊥⊥= −v .
A vector of length 1 is said to be a unit vector. A pair {v ,w} of
unit orthogonal vectors is called an orthonormal pair.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Orthonormal Pairs
Theorem
Let {v ,w} be an orthonormal pair of vectors in R2. Then for allx ∈ R2,
x = �x , v�v + �x ,w�w .
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Equation of a Line
If � is a line with direction vector v , the vector v⊥ is called a
normal vector to �.
Any two normal vectors to the same line are proportional.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Equation of a Line
Theorem
Let P be any point and let {v ,N} be an orthonormal pair ofvectors. Then
P + [v ] = {X |�X − P ,N� = 0}
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Equation of a Line
Corollary
If N is any nonzero vector, {X |�X − P ,N� = 0} is the line throughP with normal vector N and hence, direction vector N⊥.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
The Equation of a Line
Theorem
Let a, b, c ∈ R. Then {(x , y)|ax + by + c = 0} is
i. the empty set if a = b = 0 and c �= 0,
ii. the whole plane R2 if a = b = c = 0,
iii. a line with normal vector (a, b) otherwise.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Two lines � and m are said to be perpendicular if they have
orthogonal direction vectors. In this case, we write � ⊥ m. Two
segments are perpendicular if the lines on which they lie are
perpendicular.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular LinesPythagorean Theorem
Theorem
Pythagoras.Let P ,Q,R be three distinct points. Then
|R − P |2 = |Q − P |2 + |R − Q|2
if and only if QP ⊥ RQ.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Theorem
If � ⊥ m then � and m have a unique point in common.
If three or more lines all pass through a point P , we say that the
lines are concurrent. If three or more points lie on the same line,
the points are said to be collinear.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Theorem
Let X be any point and let � be a line. Then there is a unique linem through X perpendicular to �. Furthermore,
i. m = X + [N], where N is a unit normal to �;
ii. � and m intersect in the point F = X − �X − P ,N�N, where Pis any point on �;
iii. d(X ,F ) = |�X − P ,N�|.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Remark
The construction of m when � and X are given is called erecting aperpendicular to � at X if X happens to lie on �. Otherwise, it iscalled dropping a perpendicular to � from X . In this case the
unique point of intersection of � and m is called the foot of theperpendicular.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Theorem
Let � be any line, and let X be a point not on �. Let F be the footof the perpendicular from X to �. Then F is the point of � nearestX .
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Definition
The number d(X ,F ) is called the distance from the point X to
line � and is written d(X , �).
Remark
d(X , �) is the shortest distance from X to any point on �.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Corollary
Let � be a line with unit normal vector N. Let X be any point ofR2. If P is any point on �, then
d(X , �) = |�X − P ,N�|.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Perpendicular Lines
Let PQ be a segment. The line through the midpoint M of PQthat is perpendicular to PQ is called the perpendicular bisectorof PQ.
Remark
The perpendicular bisector consists precisely of all points that are
equidistant from P and Q.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Parallel and Intersecting Lines
Two distinct lines � and m are said to be parallel if they have no
point of intersection. In this case we write � � m.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Parallel and Intersecting Lines
Theorem
Two distinct lines � and m are parallel if and only if they have thesame direction.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Parallel and Intersecting Lines
Theorem
i. If � � m and m � n, then either � = n or � � n.
ii. If � � m and m ⊥ n, then either � ⊥ n.
iii. If � ⊥ n and m ⊥ n, then either � = m or � � m.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Parallel and Intersecting Lines
Theorem
Let � and m be parallel lines. Then there is a unique numberd(�,m) such that
d(X , �) = d(Y ,m) = d(�,m)
for all X ∈ m and allY ∈ �. In fact, if N is a unit normal vector to� and m for any points X on m and Y on �,
|�X − Y ,N�| = d(�,m).
Thus, parallel lines remain “equidistant.”
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
Parallel and Intersecting Lines
Theorem
Let � be any line, and let m be a line intersecting � at a point P.Let v and w be unit direction vectors of � and m, respectively. Letα(t) = P + tw be a parametrization of m. Thend(α(t), �) = |t||�w , v⊥�|. Thus, as X ranges through m, d(X , �)ranges through all nonnegative real numbers, each positive realnumber occurring twice.
Ederlina Ganatuin-Nocon MODERN GEOMETRY
Plane Coordinate Geometry
PROBLEM SET #1Date Due
To be submitted on:13 June 2013, 1700HRS
Ederlina Ganatuin-Nocon MODERN GEOMETRY
MODEGEO Problem Set #1 13 June 2013
Directions: Show your solution to each of the following.
1. If � = P + [v] = Q+ [w], how must P , Q, v and w be related?
2. If 0 < t < 1 and X = (1− t)P + tQ, and P �= Q, show that
d(P,X)
d(X,Q)=
t
1− t.
Use this to find the point X that divides the segment PQ in the ratio r : s. Illustrate using r = 2, s = 3, P =
(−3, 5), Q = (8, 4).
3. Find an orthonormal pair one of whose members is proportional to (5,−12).
4. (a) Find all unit normal vectors to the line 3x+ 3y + 10 = 0.
(b) Find all unit direction vectors of the same line.
(c) If P = (5, 2) and v = (12 ,
23 ), find the equation of the line P + [v] in the form ax+ by + c = 0.
5. If v = (v1, v2) is a direction vector of a line �, the number α = v2/v1 is called the slope of �, provided v1 �= 0.
(a) Show that the concept of slope is well-defined.
(b) Show that if � is a line with slope α, the vector (1,α) is a direction vector of �.
(c) Show that the line through P = (x1, y1) with slope α has the equation
y − y1 = α(x− x1).
6. Let P + [v] and Q+ [w] be intersecting lines. Let D be the matrix whose first row is v and whose second row is w. IfP − tv = Q+sw is the point of intersection, prove that (t, s) = (P −Q)D−1
. Here (t, s) and P −Q are regarded as 1×2
matrices. Use this method to find the intersection point in the case P = (1, 5), Q = (3, 7), v = (8, 1) and w = (6, 2).