# Vectors

Vectors have magnitude and direction.

Example: Speed has a value and a direction.

Definition

The vector

### Conventions and notation

- Two vectors with the same length and the same direction are considered to be the same.
- The name of a vector can be written in bold:
**v**or underlined v

### Standard position and component form

Defintion

- A vector is in
**standard position**if the origin is its initial point. - If the vector equals OP where O is the origin and P = (x, y, z), then we write
**v**in**component form**:

info

If vector **v** has initial point

### The length of a vector

Definition

The length of the vector v with initial point P and terminal point Q is denoted as

If

, then *(in standard position)*If

then

### Unit Vectors

Definition

A unit vector is a vector with length = 1.

- to get a unit vector from v:
, then is a unit vector

### Direction

Definition

The direction of a non-zero vector v is the vector

- Directions are unit vectors.
- If a direction is in standard position, its end point lies on the unit circle (in
), or the unit sphere (in )

### Special vectors

**zero vector**: all components are 0, notaion:**0****standard base vectors**:

### Scalar multiplication of vectors

Scalar multiplication of a vector v with a real number a results in a shorter or longer vector with the same or opposite direction as v.

info

If

Then

### Sum of two vectors

Two visual ways:

head to tail construction: shift v such that the initial point of v is the same as the terminal point of u.

parallelogram construction: shift v such that the initial points if u and v coincide.

algebraically:

info

If

Then

#### Properties

- a(u + v) = au + av
- (a + b)u = au + bu

### The difference of two vectors.

visually:

- head to head: shift the vectors such that the initial points coincide
- v - u is the vector from the head of u to the head of v.
- u - v is the vector from the head of v to the head of u.

### Dot product

Definition

The dot product of two vectors v and u:

- the dot product is a number.
- the dot product can be used to describe the length of the vector:

#### Properties

### The angle between unit vectors.

info

if u and v are unit vectors, and

info

if u and v are non-zero vectors and

### Orthogonality

Definition

two vectors u and v are orthogonal or perpendicular if

- if
, then the angle between u and v is - if u = 0 or v = 0 then there is no angle between u and v but we still say that u and v are orthogonal.
- we denote orthogonal vectors with the "
" symbol :

### Projection

in other words:

Definition

Let w and v be two vectors, **Projection** of w onto v is:

### Cross Product

- Only in 3 dimensions => vector has 3 components

Definition

Let

#### Cross Product Template:

- Write u and v in a column:
- copy the first two leftmost entries of u and v.
- calculate the first entry of u x v, starting with u_2 in an x shape.
- calculate the second entry of u x v.
- calculate the third entry of u x v.

#### Properties:

#### Geometric Properties:

with being the angle between u an v. - right hand rule for
: - index finger: first vector (u)
- middle finger: second vector (v)
- thumb: cross product (
)

## Lines and Planes

Definition

The line through P parallel to v is given by

where

- The vector p is called a support vector of l.
- the vector v is called a direction vector of l.

### Planes in

- A
**normal vector**of M is a non-zero vector orthogonal to M. - A plane is determined by:
- three points, not on one line.
- by a support vector and a normal vector.

#### Planes by normal vector:

- If n is a normal vector of M, then for every
- if
this equation is called the **normal equation**of M.

#### Planes determined by three points:

How to find a normal vector: If P, Q and T are three points (not on a line), a normal vector of the plane can be found by taking a cross product, for example:

#### Method: Finding a plane equation from three points

*The Plane W goes through three points A, B and C.*

*Determine an equation for the plane W of the form *

- Find a normal vector for the plane W.
- Find a vector:
- Find another vector:
- Find the cross product of the vectors
**AB**and**AC**(we use u and v for readability)

- Find a vector:
- Find a support vector p, which is a vector from the origin to any point on the plane.
- Choose

- Choose
- Replace n and p in
(x can be replaced with )