Find the projection of the vector $ \vec{v_1} = \left(0,~0\right) $ on the vector $ \vec{v_2} = \left(0,~0\right) $.

The projection of $ \vec{v_1} $ on the vector $ \vec{v_2} $ is given by

$$ \text{Proj}_{\vec{b}}{\vec{a}} = \dfrac{ \vec{a} \cdot \vec{b} }{ \| \vec{a} \|^2 } \vec{b}$$

First we find the dot product and magnitude of vector $ \, \vec{b} $:

$$ \begin{aligned}\vec{a} \cdot \vec{b} &= 0 \\[1 em]\| \vec{b} \| &= 0 \\[1 em]\end{aligned} $$

Now we can find the projection

$$ \text{Proj}_{\vec{b}}{\vec{a}} = \dfrac{ 0 }{ \left( 0 \right)^2 } \cdot \vec{b} = \dfrac{ 0 }{ 0 } \cdot \vec{b} = Cannot divide by zero. \cdot \left(0,~0\right) = \left(Cannot divide by zero.,~Cannot divide by zero.\right) $$

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