How do I find the angle between 2 vectors?

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First, we need to recall 2 basic definitions of vector operations:

The dot product is defined on vectors u=[u1, u2,...un] and v=[v1, v2,..., vn] as u . v = u1v1+u2v2+...+unvn
The length (norm) of a vector v=[v1, v2,..., vn] is the nonnegative scalar defined as ||v||=√(v . v)=√(v12+v22+...+vn2)

Note that u & v must be the same size to compute the dot product.

Now the formula for the angle, θ, between 2 vectors is as follows:

             cos(θ)=(u . v)/(||u|| ||v||)

Notice that u & v can be any size so long as they are both the same size. That is, this formula can be used to find the angle between vectors in 2 dimensions and also to find the angle between vectors in 100 dimensions, however hard that is to imagine.

A handy rearrangement of that formula to isolate θ is:

θ=cos-1( (u . v)/(||u|| ||v||) )


Christopher H. A Level Maths tutor, A Level Further Mathematics  tuto...

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