what does the value of the dot product of two vectors tell us?
In: 10
Depends what the vectors are.
If your vectors are force and motion, the dot product tells us the total work done. If your vectors are the electric field gradient and charge flux, it tells us the power per unit length.
Usually, dot products arise when multiplication happens but in multi-dimensional space where the things being multiplied may not be in the same direction. This actually also applies to cross products, though, so don’t treat this as an exhaustive definition or anything.
The dot products tells you **how similar** and **how big** the two vectors are.
A dot product of 0 means that either the vectors have “nothing in common”, meaning they are orthogonal to one another, or one of the vector is 0.
If two vectors go exactly in the same direction, so they are as similar as they can possibly be, then the dot product is simply the multiplication of both lengths. More generally, if you increase the length of a vector, the dot product will increase proportionally.
If you’ve done some geometry, you probably know how important **angles** and **lengths** are. The dot product is combining those two notions into a single one, and while this might sound unintuitive, it does wonders for mathematical formula.
You can visualize it as a *projection* of one of the vectors onto the other.
If you have the two vectors as arrows, then shine a light on one of the arrows. Its shadow on the other vector is equal to the dot product.
Latest Answers