# Theoretical Mechanics/Vector Algebra

## Contents

## Scalars and vectors[edit]

### Scalars[edit]

Scalars are simple values. They describe something like height, length, mass, number of chairs with a real number. e.g. 10 chairs, 5Kg, 2m: 10, 5 and 2 are scalars.

### Vectors[edit]

Vectors have a magnitude and a direction. e.g. the velocity of a ball has both a speed and a direction in which it is moving.

### Basis Vectors[edit]

All cartesian coordinate systems have a number of basis vectors equal to their dimension; for most applications in Theoretical Mechanics the vector space is E_{3}, whose basis vectors are **e _{x}**,

**e**,

_{y}**e**.

_{z}You may also see
for **e _{x}**,
for

**e**, and for

_{y}**e**.

_{z}Every vector is then a linear combination of these basis vectors. Often, vectors are written as tuples, so that the vector a**e _{x}** + b

**e**+ c

_{y}**e**is equivalent to the tuple (a, b, c).

_{z}The unity vector of any vector can be found by dividing all of its components by the length:
V = 3e_{x} + 4e_{y}

length: |V| = sqrt (3² + 4²) = sqrt (9 + 16) = sqrt 25 = 5

Unity vector e_{v} = (3/5)e_{x} + (4/5)e_{y}

3/5 equals to the cosinus of the angle of this vector with the X-axis, 4/5 is the cosinus of the angle of this vector with the Y-axis:

argcos(3/5) = about 53°

argcos(4/5) = about 37°

Notice that both angles add up to 90°. This means this vector lays in the XY-plane. If it adds up to a larger number, it lays inside a 3D volume, not a plane.

#### Equality[edit]

Two vectors are equal if and only if both their magnitude and their direction are the same.

4**e _{x}** ≠ 5

**e**

_{x}4

**e**≠ 4

_{x}**e**

_{y}4

**e**+ 8

_{x}**e**- 1

_{y}**e**≠ 4

_{z}**e**+ 8

_{x}**e**+ 3

_{y}**e**

_{z}4

**e**= 4

_{x}**e**

_{x}10

**e**= 10

_{y}**e**

_{y}4

**e**+ 8

_{x}**e**- 1

_{y}**e**= 4

_{z}**e**+ 8

_{x}**e**- 1

_{y}**e**

_{z}#### Vector multiplied with scalar[edit]

Multiply the magnitude of the vector with the scalar.

**4**e_{x} + **3**e_{y} *** 2** = **8**e_{x} + **6**e_{y}

3e_{a} * 5 = 15e_{a}

#### Angle between 2 vectors[edit]

This is the smallest angle between both vectors.

#### Adding Two Vectors[edit]

Two vectors are added component by component, like this:

(a_{1}**e _{x}** + a

_{2}

**e**+ a

_{y}_{3}

**e**) + (b

_{z}_{1}

**e**+ b

_{x}_{2}

**e**+ b

_{y}_{3}

**e**) = (a

_{z}_{1}+ b

_{1})

**e**+ (a

_{x}_{2}+ b

_{2})

**e**+ (a

_{y}_{3}+ b

_{3})

**e**

_{z}So, for example:

(2**e _{x}** + 1

**e**+ 3

_{y}**e**) + (

_{z}**e**+ 1

_{x}**e**+

_{y}**e**) = (3

_{z}**e**+ 2

_{x}**e**+ 4

_{y}**e**)

_{z}#### Subtracting Two Vectors[edit]

Subtracting a vector **A** from a vector **B** is no different than adding **A** and -1 * **B** together.

#### Dot product[edit]

a*b*cos θ

With: a, b magnitude of the vectors; θ angle between the vectors.

A.B = a_{x} * b_{x} + a_{y} * b_{y} + a_{z} * b_{z}

with A,B 2 vectors; a_{x} X-component of vector A, ...

Properties of the dot product: Is a scalar. When equal to zero: The angle between the vectors is 90°. or one of the vectors is zero. Maximum: angle between the vectors is 0°

#### Cross product[edit]

U = A x B

U = a * b * sin θ

U = (A_{y} * b_{z} - a_{z} * b_{y}) e_{x}

+ (A_{z} * b_{x} - a_{x} * b_{z}) e_{y}

+ (A_{x} * b_{y} - a_{y} * b_{x}) e_{z}

Properties of the cross product: Is a vector orthogonal to both A and B