**Section 1.1**

**Vectors in Euclidean Spaces**

**Introduction**

__Def.__
*R*
Set of all real numbers. Also called
*scalars.*

xy-plane, ordered pairs (a,b).

space, ordered triples (a,b,c).

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

be regarded as a point
*x*
=(
,...,
) and represented graphically as

a dot.

**Vectors **

A
*vector*
in the Euclidean space
is an array consisting of n components,
*x*
=(
,...,
).

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

include "position".

Vector in 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]);**

**(1)**
*v *
is a vector of three components:

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

`> `

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

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

`> `

**(3) **
*w*
is a vector with five equal entries

`> `
**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]);**

`> `

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);**

`> `

Solve the resulting equations:

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

`> `

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

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

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

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

Addition

**Example 1.2**
Consider the two vectors v and u in

`> `

`> `
**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);**

`> `

`> `

`> `

**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);**

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 . Therefore:

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

**1**
. T
he set of vectors is closed under addition

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

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

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

`> `

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

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

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

This result is true in .

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

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

`> `

`> `
**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));**

`> `

`> `

and

`> `

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

`> `

`> `

`> `

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

`> `

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

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

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

This result is true in .

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

Add, say the vector w to the zero vector

`> `
**z:=vector(2,0);**

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

`> `

What does this tell you?

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

** 4**
.
The zero vector is an additive identity

This result is true in .

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

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);**

`> `

`> `

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

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

This result is true in .

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

**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]);**

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

this can be displayed as follows.

`> `

`> `
**plotvectorscalar(3,v);**

`> `

`> `

`> `

`> `
**plotvectorscalar(-3,v);**

`> `

`> `

`> `
**plotvectorscalar(1/2,v);**

`> `

`> `

`> `
**plotvectorscalar(1,v);**

`> `

`> `

Algebraically,

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

`> `

Check if the
**distributive properties**
hold

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

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

`> `

Comparing *v + *v and ( + )*v, we have

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

** **
**6.**
(
+
) * v =
*v +
*v
for any scalars
,
and any vector v

This result is true in .

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

Compare the following,

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

`> `

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

`> `

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

** 7**
.
*(v + u) =
*v +
*u for any scalar
and vectors v and u

This result is true in .

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

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));**

`> `

We conclude

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

** 8**
.
(
*
)*v =
*(
*v) =
*(
* v)

This result is true in .

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

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

`> `

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

** 9**
.
1*v = v for any vector v

This result is true in .

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

**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
,
,...,
in
is a vector of the form

*+*
+. . . +
where each
is a scalar. Every vector

in
can be expressed uniquely as a linear combination of
*standard basis*

vectors , , . . ., where has a 1 as its ith component and zeros

for all other components.

**Example 1.1**

Can the vector

`> `
**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]);**

That is, is w =
* *
?

`> `
**plotplane(w,v,u);**

Evaluate

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

For the vector w to be equal to

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

we need to check if there are values and that satisfy the set of equations:

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

`> `

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

**Example 1.2**

Can the vector

`> `
**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]);**

`> `
**plotplane(w,v,u);**

`> `

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

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

NULL

`> `

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

In general, a vector w is a
**linear combination**
of a set of vectors
....
if one can

find scalars .... such that : w:= +........+

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

The set of all linear combinations S of a given set of vectors
* *
= {
,
,
....
}

is defined by

S:= { +........+ | , , ,..., are scalars}.

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

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

**Transpose**

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

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