# Vectors

The equation of a line can be represented by vectors in 3D. The line that passes through **A** and **R** can be written as:

r=a+ λb

where

__a__is the position vector of a known point on the line__b__is the**direction vector**of the line (a vector parallel to the line)__r__is the position vector of any arbitrary point on the lineλ is a scalar parameter

If you need to find the equation of a line from just two points, **C** and **D**, and you do not know the direction vector, use the fact that the vector between the two points (__d__ - __c__) is the direction vector of the line:

The equation of this line is given as

r=c+ λ(d-c)

### Equations of 3D line in Cartesian Form

If you need to find the equation of a straight three dimensional line in Cartesian form (in terms of x, y and x), convert using the vectors __a__ and __b__ from the vector equation __r__ = __a__ + λ__b.__

## Equations of a Plane in 3D

The vector equation of a plane is given as:

r=a+ λb+ μc

where

__a__is the position vector of a known point, A, in the plane__b__and__c__are non-parallel, non-zero vectors in the plane__r__is the position vector of any arbitrary point, R, in the planeλ and μ are scalar parameters

### Normal Vector

The **normal vector** of a plane is used to describe the direction of the plane. It is the vector that is exactly perpendicular to the plane.

### Equation of a 3D plane in Cartesian Form

You can use the normal vector of a plane to write a Cartesian equation describing the plane.

Where the normal vector, **n** = a**i** + b**j** + c**k**, the Cartesian equation of the plane is given as:

ax + by + cz = d

a, b, c and d are all constants

## Scalar Product (a.k.a "Dot Product")

The scalar product, __a__ . __b__, is given by the magnitude of the two vectors __a__ and __b__, and by the angle between them, θ:

a.b= |a| |b| cos(θ)

If

**θ = 90°**(the two vectors are perpendicular), then the**scalar product will equal zero.**If

__a__and__b__are parallel__a__.__b__=**|**__a__| |__b__|If the two

**vectors are identical**(both are__a__), the**scalar product will equal |**__a__|²

### In Cartesian Form

### Scalar Product Equation of a Plane

The equation of a plane can also be written as the scalar product of the normal vector to the plane and the position vector of any arbitrary point on the plane:

r.n= k

__r__is the position vector of any arbitrary point on the plane__n__is the normal vector of the planek is a scalar constant for the plane, where k =

__a__.__n__for a specific point in the plane with position vector__a__