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Let $\vec{a}$ & $\vec{b}$ be two non-collinear, non-zero co-initial vectors having angle $\theta$ between them. The projection of $\vec{b}$ on $\vec{a}$ is given by the dot product of $\vec{b}$ &$\hat{a}$.

This is the mathematical definition. But what it is actually? What is the definition of it?

[ I guess it is the magnitude of component of $\vec{b}$ on $\vec{a}$.]

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Consider for example, a plane vector and two orthogonal unit vectors $\hat x$ and $\hat y$.

Any vector in the plane can be expressed as

$$\vec v = (\vec v \cdot \hat x) \;\hat x + (\vec v \cdot \hat y) \; \hat y = v_x\; \hat x + v_y\; \hat y$$

So, you're correct, $\vec b \cdot \hat a$ is the component of $\vec b$ in the $\hat a$ direction.

And further, the operator

$$\left(\quad\cdot\; \hat a\right) \hat a $$

is a projection operator - it takes as input a vector and returns a vector - the projection ('shadow') of that vector in the $\hat a$ direction.

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