What do the properties of the dot product mean?

Question

I am having trouble understanding the relevance or meaning of the properties of dot product.

For example, the distribution property of dot product states:

$$\vec a \cdot (\vec b+\vec c) = \vec a \cdot \vec b + \vec a \cdot \vec c$$

Or the Scalar Multiplication Property states:

$$(c\vec a) \cdot \vec b = \vec a \cdot (c\vec b) = c(\vec a \cdot \vec b)$$

Can anyone explain why this is important.

Answer 1

You question seems to have gotten truncated, but here's the deal:

Dot products have a formula, as you've seen in the definition. But in practice, in many of the places where we apply dot products, the actual formula doesn't matter a lot: what we need is to be able to do some algebra on expressions involving dot products.

As an analogous example, suppose I have a number $s$ that satisfies $s^2 = 5 + 2 \sqrt{2} \sqrt{3}$. I'd like to evaluate $s(s-s^3)$. The way I do this is to write $$ s(s - s^3) = s^2 - s^4 = s^2 - (s^2)^2 $$ and now I can substitute $s^2 = 5 + 2 \sqrt{2}\sqrt{3}$ to get the final result. In the first couple of steps, the actual value of $s$ didn't matter at all: all I used was properties (like distributivity and exponentiation rules) that apply to every real number.

In the same way, working with dot products, you'll often find yourself encountering something like $v \cdot (w + u)$, and want to be able to change this to $v \cdot w + v \cdot u$; the distribution property above says that it's OK to do this, without resorting to the definition of the dot product -- you don't have to write out everything in terms of sums of products of corresponding entries of your vectors. Not writing that out makes it easier to follow the big picture of whatever computation you're doing, and that's nice.

Answer 2

Well my answer is going to be in physics( since it is an intuition question) . The applications of work done by a force F. so if a force moves an object by a displacement of $a$ , then followed by a displacement of $b$, then intuitively, we can see that the work done( the energy ) is the sum of the 2 displacements. So, $$W=F•(a+b)$$ Due to distributive property of dot product, we can infer that the work done on the object is simply the sum of the work done after the first displacement and after the second displacement.mathematically, $$W=F•(a+b)=F•a+F•b=w_{1}+w_{2}$$ This is physics, and we see that the mathematical property of dot product can justify some real world physical definitions.