# Vectors

## Vector Equation of a Straight Line

The vector is the vector of any point along the line. is any point on the line, and \bm{b} is the direction of the line. is a parameter that represents the position of relative to along the line. The carteian form of this can be derived:

## Scalar/Dot Product

The dot product of two vectors:

• If , then and
• The two vectors are perpendicular

The angle between two vectors can be calculated using the dot product

## Projections

The projection of vector in the direction of is given by the scalar product:

This gives a vector in the direction of with the magnitude of .

## Equation of a Plane

The vector equation of a plane is given by

Where is the normal to the plane, and is any point in the plane. This expands to the cartesian form:

## Angle Between Planes

The angle between two planes is given by the angle between their normals.

## Intersection of 2 Planes

Two planes will only intersect if their normal vectors intersect.

• First, check the two normals are non parallel
• Equate all 3 variables about either a parameter or one of , , or to get an equation for the line along which the planes intersect in cartesian form

### Example

Find the intersection of the planes (1) and (2).

(1) - (2):

(1) + 3(2):

Equating the two with z:

### Using Cross Product

For two normals to planes and , the vector will lie in both planes. The line

lies in both planes.

## Distance from Point to Plane

The shortest distance from the point to the plane is given by:

## Vector/Cross Product

The cross product of two vectors produces another vector, and is defined as follows

is the angle between the two vectors, and is a unit vector perpendicular to both and . The right-hand rule convention dictates that should always point up (ie, if and are your fingers, then is your thumb). The cross product is not commutative, as = . • The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors.
• Can be used to find a normal given 2 vectors/2 points in a plane

### Angular Velocity

A spheroid rotates with angular velocity . A point on the spheroid has velocity