What's the dot product?
Nathan Kamgang
If you interrogate the formula for the dot product, it will gift to you all the explanations you need. You’ve seen mathematicians define the same dot product in different ways.
The geometric definition of the dot product
In the two dimensional plane, start wit two vectors at the origin. Just think of our vectors as line segment that draw starting at coordinate (0,0)–your origin. The dot product of those two line segments(vectors) equals u.v = costheta times length u times lenght v
Draw a diagram for those 2 vectors here
Why is it defined this way? Think of it as an index chosen abitrarily. Biologist have the Body Mass INdex wich is mass divided by square of height. Economist have Human development index… We define an index because it tells us something about the object The geometric dot product is just and index made of of the lenght of two line segments multiplied by the cos of the angle between them.
What does such index tell us? To know this we must observe how the object behaves. The lenght of a line segment is always positive(If you don’t believe me try to visualize a rule of -15 ) On the other hand cos can be negative. Our line segments above can be viewed as two sides of a triangle. Divide the longest vector by the smallest vector and you get the cosine. It measure. Costheta measure how our ratio of the two sides of our vector changes. Take a piece of paper and draw 2 line segments with a common origin. Draw the same line segment with the bigger angle. If you want to join our line segment to form a triangle notice how changing the angle determine changes the size of our line segment.
Insert picture with two angles with the same line segment
The range of the cosine function is -1 to 1 so the cosine can be negative.
We |u| only positive |v| only positive by the rule of multiplication. The cosine will solely determine the sign of the dot product
The lenght of the vectors multiply the cosine so the longer the lenght the. The bigger the dot product if cosine is positive Or the smaller the dot product if cosine is negative. In any case, increase the lenght and you scale the dot product.