1.Vectors in Space 2.Length and Angle Measures II. Linear Geometry of n-Space.

1. Vectors in Space

2. Length and Angle Measures

II. Linear Geometry of n-Space

II.1. Vectors in Space See Apostol, Chap 1.

Descartes: Points in plane ~ ( a1 , a2 ), points in space ~ ( a1 , a2 , a3 ).

Cayley, Grassmann: Points in n-space ~ n-tuples ~ ( a1 , a2 , …, an ) n.

Bound vector: a = ( a1 , a2 , …, an ) = arrow from origin to point a.

Vector algebra in n :For all a, b, c n and α , 1. Vector equality: a = b ai = bi i = 1, …, n.2. Vector addition: c = a + b ci = ai + bi

3. Scalar Multiplication: a = αb ai = α bi

Parallelogram rule:

Closure: n is closed under all linear combinations of its elements:i nv R i

i

i nc v R ic R n is a linear space

Free vector: Object with a magnitude & a direction.

Arrow from point to point .ab

a b

ab

b a 1 1 , , n nb a b a

2 parallel arrows of the same length represent the same free vector.

bound vector of components bi ai .

Comment:

• Free vectors can be defined only in a linear (flat) space.

• In a (non-linear) curved space, parallelism must be defined explicitly.

( Each point has its own linear vector space. )

• Riemannian geometry.

• Differential geometry.

n is isomorphic to the space Mn1 of n1 column matrices.

Definition: 2 spaces are isomorphic if there exists a bijection (1-1 onto mapping) between them that preserves the algebraic structures.

1, , nna a a R

11n

n

a

M

a

a

1 1, , nn na b a b a b R

1 11n

n n

a b

M

a b

a b

11n

n

a

M

a

a 1, , nna a a R

i ia b i a b

A real vector space is any space ( V, + ; ) isomorphic to n .

Group Isomorphism

I = [ 0, 2π) cos sin; 0,2

sin cos

RR

{ I, } is a group if we define as 2a b a b Mod

{ , } is a group if we define as the matrix multiplication

1 2 1 2 R R R R

{ I, } & { , } are isomorphic with f : I → by θ R(θ)

Let { A, *} & {B, } be 2 groups.

They are isomorphic if there exists a 1-1 onto mapping f s.t.

f(a) = b where aA, bB

Furthermore, f preserves the group operations, i.e.,

a1 * a2 = a3 f( a1 ) f( a2 ) = f( a3 )

Example

Analytic Geometry

Definition: Line

A line spanned by a is the set of points

L( 0 ; a ) = { t a | t } [ L(0 ; a ) passes through 0 ]

A line through point p & parallel to that spanned by a is the set of points

L( p ; a ) = { p + t a | t }

Corollary: A line running through points p & q is the set of points

L( p ; qp ) = { p + t (qp ) | t }

Example: Line through points (1,2) & (3,1) in 2.

1 3 1

2 1 2L t t

R

1 2

2 1t t

R

Line through points (1,2) & parallel to (2,1).

1 2

2

tt

t

R

Definition: Plane

A plane spanned by a and b is the set of points

P( 0 ; a, b ) = { s a + t b | s, t } [ P(0 ; a, b ) passes through 0 ]

A plane through point p and parallel to that spanned by a and b is the set of points

P( p ; a, b ) = { p + s a + t b | s, t }

Corollary: A plane running through points p , q & r is the set of points

P( p ; qp , rp ) = { p + s (qp ) + t (rp ) | s, t }

Example: Plane through points (1, 0, 5), (2, 1, 3), & (2, 4, 0.5) in 3.

1 2 1 2 1

0 1 0 4 0 ,

5 3 5 0.5 5

P s t s t

R

1 1 3

0 1 4 ,

5 8 4.5

s t s t

R

Example: Plane given by 2x + y + z = 4 in 3.

Same as plane through points (2, 0, 0), (0, 4, 0), & (0, 0, 4) in 3.

2 0 2 0 2

0 4 0 0 0 ,

0 0 0 4 0

P s t s t

R

2 2 2

0 4 0 ,

0 0 4

s t s t

R

2 1 1

0 2 0 ,

0 0 2

s t s t

R

Exercises 1.II.1.

1. (a) Describe the plane through (1, 1, 5,1), (2, 2, 2, 0), and (3, 1, 0, 4). (b) Is the origin in that plane?2. A person traveling eastward at a rate of 3 miles per hour finds that the wind

appears to blow directly from the north. On doubling his speed it appears to come from the north east. What was the wind’s velocity?

3. Intersect these planes:

1 0

1 1 ,

1 3

s t s t

R

1 0 2

1 3 0 ,

0 0 4

m n m n

R

II.2. Length and Angle Measures

Definition 2.1: Euclidean norm (Length)

The Euclidean norm || v || of a vector v n is2 21 nv v v

1/ 2

2

1

n

ii

v

0

Law of Cosines: 2 2 2 2 cosa b c b c

Proof: Pythagorean theorem:

2 22 sin cosa b b c

2 2 2 2sin cos 2 cosb c b c

2 2 2 cosb c b c

Example: Angle between 2 vectors.

Law of Cosines: 2 2 22 cos u v u v u v

2 2 2

1 1

2 cosn n

i i i ii i

u v u v

u v

1

cosn

i ii

u v

u v

11cos

n

i ii

u v

u v

11

2 2

1 1

cos

n

i ii

n n

i ji j

u v

u v

Definition 2.3: Dot/Inner/Scalar Product

1

n

i ii

u v

u v 1 1 n nu v u v

2

1

n

ii

u

u u2 u

1cosu v

u v

2 2 2 cosu v u v u v u v

2u

Law of Cosines:

cosu v u v

1cos

u v

u v

v u

Exercise: Show that

a a a u v u v u v & , na u vR R

( Definition 2.7 )

Theorem 2.5: Triangle Inequality

u v u v u v

Shortest path between 2 points is a straight line.

Geodesics in a Euclidean space are a straight lines.

The sum of 2 edges of a triangle is greater than or equal to its 3rd edge.

Proof: Since all norms are positive, the inequality is equivalent to

22 2u v u v u v

or 2 2u v u v u v u v v u

uv u v ( Cauchy-Schwarz Inequality ; to be proved )

cos 1

Setting v → v gives uv u v cos 1 → cos 1 →

Corollary:

, n u v R

2 2 2u v uv

Theorem 2.6: Cauchy-Schwarz Inequality uv u v u v

Proof:

Let a b z u v where ,a bR

2z a b a b u v u v 2 2 2 2 2a u b v ab u v 0

Let 2a v b u v&

→ 2 24 2 2 22 0v u v v u v u v

22 2v u u v

uv u v

a, b

, n u v R

Definition: Vectors u, v are orthogonal if u · v = 0.

Conclusion: ||w|| 0 u v u v u v u v

Exercises 1.II.2.