Understanding Dot and Cross Product

When you deal with vectors, sometimes you say to yourself, "Darn I wish there was a function that..."

  • was zero when two vectors are perpendicular, letting me test perpendicularness."

    Dot Product

  • would let me find the angle between two vectors."

    Dot Product (actually gives the cosine of the angle between two normalized vectors)

  • would let me 'project' one vector onto another, or give the length of one vector in the direction of another."

    Dot Product

  • could tell me how much force is actually helping the object move, when pushing at an angle."

    Dot Product

  • could tell me how much a vector field is 'spreading out'."

    Cross Product

  • could give me a vector that is perpendicular to two other vectors."

    Cross Product

  • could tell me how much torque a force was applying to a rotating system."

    Cross Product

  • could tell me how much this vector field is 'curling' up."

    Cross Product

There are actually a lot more uses, but the more I study vectors, the more and more I run into a situation where I need a function to do exactly something, and then realize that the cross/dot products already did exactly what I needed!


The dot product can be used to find the length of a vector or the angle between two vectors.

The cross product is used to find a vector which is perpendicular to the plane spanned by two vectors.


Two more uses that I haven't seen mentioned yet: if you want to find the area of the parallelogram formed by two vectors (each vector gives a pair of parallel sides), then you would use the magnitude of the cross product of the two vectors.

One use of this is to aid in defining a surface integral. Let x(u,v) be a parameterization of a surface. Then at each point, we can find tangent vectors Tu = ∂x/∂u and Tv = ∂x/∂v. From the idea of a linear approximation, Tu and Tv will define a tangent plane at that particular point. Consider the parallelogram formed with Tu and Tv as sides. Informally, we can see that each area element will be |Tu x Tv| du dv. Then a function f(u,v) integrated over this surface is ∫∫ f(u,v) |Tu x Tv| du dv.

As for the second use, if you wanted to find the volume of the parallelepiped having the three vectors a, b, c as sides, then you would use the magnitude of the scalar triple product |a ⋅ (b x c)|.