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For $$\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$$ and $$\vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k}$$:
$$\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$$
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For $$\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$$ and $$\vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k}$$:
$$\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$$
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