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What gives us the product of 2 vectors in 2-dimensional space? Just please explain the nature and significance (why and how to use it in practice) on the use of plain language, and not just a "scalar" for example). Thank you.
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The dot product gives the projection of one vector on another. In other words, you get the length of one vector in the direction of the other.

For example, you're taking the trolley. Trolley of mass 1 kg moving forward with acceleration a, and you pull it at an angle of 30 degrees to the ground with a force f = 10 Newtons. So, truck, roughly speaking, is the acceleration of 'a*f=|'a|*|f|*Sin(30) = 5 Newtons.
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For normalized vectors (have length 1) reflects the angle between the vectors (the inverse cosine). Got 0 the vectors are perpendicular, 1 - look in one direction and -1 the opposite.
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In 2D there are two products of vectors (x,y) and (x',y').
• Scalar xx' + yy'. Projection one to the other, multiplied by the length of the other. Equal to zero, if the vectors perpendicularly.
• Oblique xy' − x'y. The area of a parallelogram of these vectors. Is zero if the vectors are collinear.
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