Vector Geometry

The Cross Product

What is the cross product?

The cross product, also known as the vector product, of two vectors gives a vector perpendicular to both the vectors, in a "right hand set". This can be very useful in defining planes etc. The cross product of two vectors is given by the following:

You can use your right hand with your thumb, index finger and second finger to see in which direction the cross product of two vectors point:

  • Use your index finger to represent the first vector (a)
  • Use your second finger to represent the second vector (b)
  • Your thumb will be pointing in the direction of the cross product of the two vectors (axb)


Calculating the Cross Product of 2 vectors

Calculating the cross product is simple. If you have experience with working out the determinant of 3x3 matrices, it will seem very familiar. The cross product of two vectors can be expressed as the following:
This notation says that the cross product is equal to the determinant of the 3x3 matrix shown above, if the matrix is set up as above. Therefore:
Why is this how you calculate the cross product? You can check out the derivation by clicking the button below:

Cross Product Derivation


Properties of the Cross product operation

1. The Cross product is anti-commutative

The cross product is an anti-commutative operation. This means that:

2. The Cross product of parallel vectors is zero

The Magnitude of the cross product of two vectors is proportional to the sign of the angle between them. If they are parallel the angle between them is 0, therefore since sin(0)=0, the cross product will be equal to 0: The cross product is an anti-commutative operation. This means that:

3. Multiplication by scalars

Because you are able to describe the magnitude of a scalar-vector product in the following way:

You can therefore state the following for the cross product of two vectors with scalar coefficients, since the direction of the vectors (thus the direction of the vector perpendicular to both of them) hasn't changed:

4. The distributive property of the cross product over vector addition

Just as scalar multiplication is distributive over scalar addition, the cross product is distributive over vector addition. What does this mean? This means that if you crossing a vector by a sum of two other vectors, you can separate the calculation into two:


Finding the Intersection of two Planes

The intersection of two planes will always be a straight line. Get two flat objects and test that for yourself! There are three different method for doing this, which suit different people.

Method 1: Find two points

Let's say, for example, that you want to find the line of intersection between the following two planes:
This first method involves finding two points which lie on the intersection, and then using these two points to define the line. This takes advantage of the fact that the intersection line is the only place where you will find common points between two planes. The first step to to arbitrarily pick one of the variables to define as a constant. For example y=0:
Solving these simultaneous equations gives the corresponding x and z points for y=0, which lie in both planes, giving us one point on the intersection line- let's call this point P:
To find the second point, similarly, again we chose to set one of the variables to a constant. This time let's chose z=1:
Solve these again, to find another point on the intersection line- let's call this one Q:
Now we have two points, on a line that we want to define. Finding the vector equation of a line is covered in depth here. To summarise, we first find the direction of the vector between the two points for example P-Q:
Then we select the first point (in this case P) to be the starting point of our line, giving us our line:

Method 2: Crossing the normal vectors


Method 3: Substituting a parameter


Finding the Angle between two Planes

Finding the Intersection of Lines

Calculating the Distance between a Point and a Line

Calculating the shortest Distance between two Skew Lines

In three dimensions lines which don't cross are called skew lines. Here we want to work out the shortest distance between two skew lines. We can think about the two lines as being in a plane and construct the following to help visualise the situation:
The distance PQ is clearly the shortest distance, and is perpendicular to both lines. This means that taking the cross product of the direction vectors of both lines will give us the direction of the shortest distance line (PQ).
Now we have the direction of the shortest distance line, we need to find its magnitude. From the vector equation of a line, you have a point on each line (the starting point).
Theses points are illustrated on the diagram below, and it is clear that the distance between the point A1 and the point B is equal to (PQ) therefore is also equal to the shortest distance we want to find.
The vectors A1A2 and A1B are base-to-base therefore the dot product will give us the following, remembering that A1B is in the same direction as the normal n we previously worked out:
From simple trigonometry it is clear that the shortest distance we want is equal to |A1A2|cosθ.
Therefore dividing the dot product we carried out in the previous step, by the magnitude of the normal vector worked out in the first step, we can get |A1A2|cosθ, thus the shortest distance between the two skew lines!

Calculating the Distance between a Point and a Plane

The Scalar Triple Product

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