Section 1.1

Vectors in Euclidean Spaces

Introduction

Def. R Set of all real numbers. Also called scalars.

R^2 xy-plane, ordered pairs (a,b).

R^3 space, ordered triples (a,b,c).

Euclidean n-spaces R^n consists of all ordered n-tuples of real numbers. Each n-tuple x can

be regarded as a point x =( x[1], x[2] ,..., x[n] ) and represented graphically as

a dot.

Vectors

A vector in the Euclidean space R^n is an array consisting of n components, x =( x[1], x[2] ,..., x[n] ).

A vector is a geometric concept that includes "direction" and "length", but does not logically

include "position".

Vector in R^n can be defined as (terminal point) - (initial point). For convenience we will place

the initial point at the origin. We say that a vector is in standard position if it starts at the origin.

> plotposition([5,3],[3,1]);

[Maple Plot]

(1) v is a vector of three components:

> v := vector([1,5,2]);

v := vector([1, 5, 2])

>

(2) u is a vector of four equal entries (i.e. zero vector):

> u := vector([0,0,0,0]);

u := vector([0, 0, 0, 0])

>

(3) w is a vector with five equal entries

> w := vector([1,1,1,1,1]);

w := vector([1, 1, 1, 1, 1])

>

What do vectors represent geometrically ? Physical quantities such as force, velocity,

weight, and acceleration are examples of vectors. Vectors are characterized by :

(i) point of application (ii) direction and (iii) "length" or "magnitude". Quantities that are only

characterized by "magnitude" such as speed and mass represent scalars.

Algebra of Vectors

Let us investigate the properties of the structure consisting of all vectors under an

appropriate definition of the operations addition and multiplication by a scalar.

Equality

Example 1.1 Consider the two vectors

> v1:=vector([2,x-y,z,x+z]);
v2:=vector([z,-2,2*x,3]);

v1 := vector([2, x-y, z, x+z])

v2 := vector([z, -2, 2*x, 3])

>

Set up the equations that will make the components of the two vectors equal

> eq1:=2=z: eq2:=x-y=-2:
eq3:=z=2*x: eq4:=x+z=3:
print(eq1,eq2,eq3,eq4);

2 = z, x-y = -2, z = 2*x, x+z = 3

>

Solve the resulting equations:

> solve({eq1,eq2,eq3,eq4}, {x,y,z});

{x = 1, y = 3, z = 2}

>

Thus the two vectors are equal provided x =1, y = 3 and z = 2.

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Two vectors are equal if and only if their corresponding components are equal.

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Addition

Example 1.2 Consider the two vectors v and u in R^2

>

> v:=vector([1,3]);
u:=vector([4,5]);

v := vector([1, 3])

u := vector([4, 5])

Geometrically, the sum of the vectors v and u can be displayed using the "graphvectadd" function.

>

> plotvectoradd(v,u);

[Maple Plot]

>

>

>

Physically

This sum represents the resultant of two forces v and u acting on a moving particle.

Algebraically

the sum of the two vectors v and u is obtained using:

> `v+u` =evalm(v+u);

`v+u` = vector([5, 8])

The components of the vector v + u are the sum of the corresponding components of

the given vectors v and u. The sum is also a vector in R^2 . Therefore:

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1 . T he set of vectors is closed under addition

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What if we add the vector u to the vector v?

> `u+v` =evalm(u+v);

`u+v` = vector([5, 8])

>

Are the two vectors v + u and u + v equal ?

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2 . The commutative property u + v = v + u holds for any vectors u and v

This result is true in R^n .

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Does the associative property hold? Choose any three vectors v, u and w in R^2

>

> v:=vector([2,3]);
u:=vector([4,1]);
w:=vector([1,5]);

v := vector([2, 3])

u := vector([4, 1])

w := vector([1, 5])

>

Geometrically , the sum of the vectors v + (u + w) and the (v + u) + w can be displayed

using the "graphvectadd" function

>

> plotvectoradd(v,(u,w));

[Maple Plot]

>

>

and

>

> plotvectoradd((v,u),w);

[Maple Plot]

>

>

>

From the graph, it seems that the two sums v + (u + w) and the (v + u) + w are equal.

Algebraically, the sums (v + u) + w and v + (u + w) are

> `(v + u) + w` =evalm(evalm(v+u)+w);

`(v + u) + w` = vector([7, 9])

> `v + (u + w)`=evalm(v+evalm(u+w));

`v + (u + w)` = vector([7, 9])

>

The two vectors (v + u) + w and v + (u + w) are equal.

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3 . The associative property (v + u )+ w = v + ( u + w) holds

This result is true in R^n .

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Add, say the vector w to the zero vector

> z:=vector(2,0);

z := vector([0, 0])

> `w+z`=evalm(w+z);

`w+z` = vector([1, 5])

>

What does this tell you?

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4 . The zero vector is an additive identity

This result is true in R^n .

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What if you add , say, the vector v to the vector (-v)?

( Geometrically , this vector is opposite in direction to v).

>

> `v + (-v)` = evalm(v+(-1)*v);

`v + (-v)` = 0

>

>

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5 . (-v) is the additive inverse of the vector v and (-v) + v = v + (-v) = 0

This result is true in R^n .

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Scalar multiplication

When we multiply a real scalar by a vector each component

of the vector is multiplied by the scalar. The result is a vector in the same direction of the

given vector if the scalar is positive and in the opposite direction if the scalar is negative.

Example 1.3 Consider the vector

> v:=vector([5,7]);

v := vector([5, 7])

What is the effect of multiply the vector v by scalar k = 3, -3, 1/2 , 1? Geometrically,

this can be displayed as follows.

>

> plotvectorscalar(3,v);

[Maple Plot]

>

>

>

> plotvectorscalar(-3,v);

[Maple Plot]

>

>

> plotvectorscalar(1/2,v);

[Maple Plot]

>

>

> plotvectorscalar(1,v);

[Maple Plot]

>

>

Algebraically,

> `k1*v` = evalm(k1*v);

`k1*v` = vector([5*k1, 7*k1])

>

Check if the distributive properties hold

> `k1*v + k2*v` =evalm(k2*v + k1*v);

`k1*v + k2*v` = vector([5*k2+5*k1, 7*k2+7*k1])

> `(k1+k2)*v`=evalm((k1+k2)*v);

`(k1+k2)*v` = vector([5*k2+5*k1, 7*k2+7*k1])

>

Comparing k[1] *v + k[2] *v and ( k[1] + k[2] )*v, we have

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6. ( k[1] + k[2] ) * v = k[1] *v + k[2] *v for any scalars k[1] , k[2] and any vector v

This result is true in R^n .

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Compare the following,

> `k1*(v + u)`=evalm(k1*(v+u));

`k1*(v + u)` = vector([9*k1, 8*k1])

>

> `k1*v + k1*u` =evalm(k1*v+k1*u);

`k1*v + k1*u` = vector([9*k1, 8*k1])

>

***************************************************************

7 . k[1] *(v + u) = k[1] *v + k[1] *u for any scalar k[1] and vectors v and u

This result is true in R^n .

***************************************************************

Now, check if the associative property holds

> `(k1*k2)*v`=evalm((k1*k2)*v);
`k1*(k2*v)`=evalm(k1*(k2*v)); `k2*(k1*v)`=evalm(k2*(k1*v));

`(k1*k2)*v` = vector([5*k1*k2, 7*k1*k2])

`k1*(k2*v)` = vector([5*k1*k2, 7*k1*k2])

`k2*(k1*v)` = vector([5*k1*k2, 7*k1*k2])

>

We conclude

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8 . ( k[1] * k[2] )*v = k[1] *( k[2] *v) = k[2] *( k[1] * v)

This result is true in R^n .

****************************************************************

> `1*v`=evalm(1*v);

`1*v` = vector([5, 7])

>

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9 . 1*v = v for any vector v

This result is true in R^n .

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Parallel Vectors

Vectors v and w are parallel ( v || w ) if they one is a scalar multiple of the other,

i.e. v = r w, where r is a scalar.

Linear Combinations

A linear combination of vectors v_1 , v_2 ,..., v_k in R^n is a vector of the form

r_1v_1 + r_2v_2 +. . . + r_kv_k where each r_i is a scalar. Every vector

in R^n can be expressed uniquely as a linear combination of standard basis

vectors e_1 , e_2 , . . ., e_n where e_i has a 1 as its ith component and zeros

for all other components.

Example 1.1

Can the vector

> w:=vector([1,0,-2]);

w := vector([1, 0, -2])

be written as a sum of scalar multiples of the vectors v and u?

> v:=vector([-4,3,8]); u:=vector([2,5,-4]);

v := vector([-4, 3, 8])

u := vector([2, 5, -4])

That is, is w = c[1]*v+c[2]*u ?

> plotplane(w,v,u);

[Maple Plot]

Evaluate c[1]*v+c[2]*u

> `c1*v+c2*u`= c1*evalm(v)+c2*evalm(u);

`c1*v+c2*u` = c1*vector([-4, 3, 8])+c2*vector([2, 5...

For the vector w to be equal to c[1]*v+c[2]*u

> evalm(c1*v+c2*u)=evalm(w);

vector([-4*c1+2*c2, 3*c1+5*c2, 8*c1-4*c2]) = vector...

we need to check if there are values c[1] and c[2] that satisfy the set of equations:

> eq1:=-4*c1+2*c2=1; eq2:=3*c1+5*c2=0; eq3:=8*c1-4*c2=-2;

eq1 := -4*c1+2*c2 = 1

eq2 := 3*c1+5*c2 = 0

eq3 := 8*c1-4*c2 = -2

>

> solve({eq1,eq2,eq3}, {c1,c2});

{c2 = 3/26, c1 = -5/26}

Example 1.2

Can the vector

> w:=vector([1,0,-2]);

w := vector([1, 0, -2])

be written as a sum of scalar multiples of the vectors v and u?

> v:=vector([-4,3,8]); u:=vector([3,1,1]);

v := vector([-4, 3, 8])

u := vector([3, 1, 1])

> plotplane(w,v,u);

[Maple Plot]

>

> eq1:=-4*c1+3*c2=1; eq2:=3*c1+c2=0; eq3:=8*c1+c2=-2;

eq1 := -4*c1+3*c2 = 1

eq2 := 3*c1+c2 = 0

eq3 := 8*c1+c2 = -2

> lprint(solve({eq1,eq2,eq3}, {c1,c2}));

NULL

>

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In general, a vector w is a linear combination of a set of vectors v[1], v[2], v[3] .... v[n] if one can

find scalars c[1], c[2] .... c[n] such that : w:= c[1]*v[1]+c[2]*v[2] +........+ c[n]*v[n]

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The set of all linear combinations S of a given set of vectors S[1] = { v[1] , v[2] , v[3] .... v[n] }

is defined by

S:= { c[1]*v[1]+c[2]*v[2] +........+ c[n]*v[n] | c[1] , c[2] , c[3] ,..., c[n] are scalars}.

S is called the span of the set S[1] . We denote this by S = span( S[1] ) or sp( S[1] ).

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Transpose

The transpose of a row vector v is a column vector and is denoted v^T .

Similarly, the transpose of a column vector is a row vector.

Exercises Section 1.1 (pp. 15-17)

1, 5, 9, 13, 15, 17, 21, 23, 25, 33, 35, 39