As you have 3 components we are talking about vectors in the 3D space (Why?)...
To calculate the angle you have to know two things:
1. Dot product of two vectors...
$\vec{a} \cdot \vec{b}$
Which is
$(a_{x} + a_{y} + a_{z}) \times (b_{x} + b_{y} + b_{z})$
2. The magnitude of the vectors...
$\parallel \vec{a} \parallel$
Which is
$\sqrt{a_{x}^{2} + a_{y}^{2} + a_{z}^{2}}$
Then you can compute the cosinus of the angle, which is reversible...
$\cos(\theta) = \frac{\vec{a} \cdot \vec{b}} { ||\vec{a}|| \times ||\vec{b}||}$
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In 2D space it is much more simple:
$\cos(\theta) = \vec{a} \cdot \vec{b}$
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