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After seeing the answer to this question by @Vijay P R I. It was my understanding that the dot product $\vec a \cdot \vec b=|\vec a||\vec b|\cos\theta$ will always return the same result regardless of whether we choose to project $\vec a$ onto $\vec b$ or $\vec b$ onto $\vec a$.

Projection of b onto a

What happens if the projection of one of the vectors is longer than the other vector itself? Is the projection of $\vec b$ onto $\vec a$ still $|\vec b|\cos\theta$?

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Yes. Because the projection of $\mathbf b$ is not onto $\mathbf a$ but on the direction of $\mathbf a$ ( you can also think to an angle $\theta >90°$).

Emilio Novati
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