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


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.



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).



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



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 .




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.




Remark

a) R2 is a vector space.

b) �·� is bilinear, symmetric, and nondegenerate.




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.




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




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).




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).



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 .



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)



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}.



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)



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



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.



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.



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).



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).



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.



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.



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 .



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.




Theorem

Let P be any point and let {v ,N} be an orthonormal pair ofvectors. Then

P + [v ] = {X |�X − P ,N� = 0}




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⊥.




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.



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.



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.



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.



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�|.



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.



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 .



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 �.



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�|.



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.



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.




Theorem

Two distinct lines � and m are parallel if and only if they have thesame direction.




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.




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.”




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.



PROBLEM SET #1Date Due

To be submitted on:13 June 2013, 1700HRS


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).

modern geometry · 2018. 9. 19. · modern geometry plane coordinate geometry ederlina...

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