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I just learned about the definition of the vector dot product and cross product.
Almost every source says that the dot product is the:
Product of the $\cos(\theta)$ component of a vector along another vector times the magnitude of the other vector.
While cross product is the:
Area of the parallelogram formed by the two vectors to be crossed.
There seems to be no meaning of what is done and why it is done so. Why do we need to find the area of the parallelogram formed when two vectors are cross produced?
It may seem as a weird question for people who have had a long bond with vectors but this question would have surely popped in the minds of everyone who is beginning the journey of vectors!
What I would like to know is the physical and a more intuitive meaning of the dot and cross products.