What does the dot product of two vectors actually represent intuitively? What is its true meaning conceptually? [closed]

Question

We define the dot product of two vectors (A and B) as : a * b * cosθ which usually just says that it is the projection of A vector on B vector multiplied to give a scalar value.

Although, I am not able to digest this idea completely and have these doubts:

• What does it actually mean to multiply two vectors? (Like am I saying that multiply something in north direction with something in east?) Also, how is it that two vectors multiply with each other to give something that is not vector? (What happens to their directions?) "It means a vector added to itself b vector times which somehow results in a number" which doesn't make much sense right?

• Now if we view the dot product as similarity between two vectors, then the part of cos θ in its formula -> a * b * cosθ makes sense but then again what does it mean to multiply the magnitudes of those vectors? What am I getting from multiplying the magnitudes of those two vectors?

• What does the number(scalar) we get after multiplying(dotting) the two vectors together represent? What information about the vectors is that number giving us and how is it even useful?

PS - It would be helpful if you were to not answer using just formulas and numbers and rather intuition for why this could/is true?

Answer 1

Suppose you start with rational numbers. You understand the idea of a number as the ratio of two integers. You understand arithmetic and all is well. Now somebody introduces you to $\sqrt2$. Your idea of number no longer works. You need to think about numbers in a new way.

You understand multiplication as repeated addition. Now here are vectors, and you need a new way to think about it.

Multiplication is an operation that takes two numbers and gives you back another number. You have certain expectations of this operation, such as following the associative law, being commutative, following the distributive law, etc.

Suppose you have a similar operation that takes two vectors. Do you get back a vector or a number? You can explore both and come up with multiplications that work. One is the dot product. The other is the cross product.

Mathematicians think this way. They look for a set of simple rules that capture the idea of multiplication in a self consistent way. They apply these rules to different objects or add more rules to see what they can come up with. They come up with beautiful abstract structures of thought. If you add up enough simple ideas, you come up with really big structures.

So you are very right to ask what is the idea they have captured? How is it like repeated addition? How is it bigger than that and yet still the same?

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Mathematicians are perfectly content if math is useless. But physicists find that mathematicians capture the ideas that govern the behavior of the universe. Distance is a vector. Force is a vector. The dot product gives you the work done.

Thinking about the physics ideas that are modeled by mathematics is a second way to get at what does multiplication mean. What structure of thought do we need to correctly model physics?

So again you are very right to ask how is this useful.

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One place you can start is with a unit vector, $\vec u$. What do you get when you take a bunch of different vectors and dot them with $\vec u$? You find you get the projection of the vectors on $\vec u$. You get the component of the vector in the direction of $\vec u$. This is an important part of the idea.

You can also gain insight by thinking about throwing a rock and $W = \vec F \cdot \vec d$. The rock gains kinetic energy if it falls downward. The gain is proportional to the distance downward. The sideways distance does not matter at all. It loses energy if it moves upward. The gain or loss is also proportional to the strength of gravity.

Answer 2

scalar product is "apparatus" that takes the components of two vectors$~\vec a~,\vec b~$ and produce scalar. The equation for 2D space is

$$\text{scalar}:= a_x\,b_x+a_y\,b_y=a\,b\cos(\phi)$$

where $~\phi~$ is the angle between $~\vec a~$ and $~\vec b~$ and $~a~,b~$ are the magnitutes of the vectors. $~\left(a=\sqrt{a_x^2+a_y^2}~~,b=\sqrt{b_x^2+b_y^2}\right)~$

Examples

• if you know the vector components you can obtain the angle $~\phi~$ $\cos(\phi)=\frac{a_x\,b_x+a_y\,b_y}{a\,b}$
• if $\vec b~$ is unit vector then the scalar is $s=a\,\cos(\phi)~$ vector $~\vec a~$ projected on vector $~\hat b~$
• if $~\vec b=\vec a~$ then the scalar is $~s=a^2~$
• if vector $~\vec a~$ is the force vector $~\vec F~$ and vector $~\vec b~$ is the path $~\vec d~$ then the scalar ,is the work $~W~$ that done by the force along the path $~W=\vec F\cdot\vec d~$
• The engine power is $~P=\vec\tau\cdot\vec \omega~$ where $~\vec\tau~$ is the torque and $~\vec\omega~$ is the angular velocity

Answer 3

First understand components of vectors.

for $\textbf{a}.\textbf{b}$

this is equal to either:

the (scalar) size of the component (or projection) of $\textbf{a}$ in the direction of $\textbf{b}$

or:

the (scalar) size of the component (or projection) of $\textbf{b}$ in the direction of $\textbf{a}$.

(Thankfully these two interpretations give the same value for the dot product).

See: here for a diagrammatic explanation and here for some example calculations.

What is the point? Two non-zero vectors are orthogonal / perpendicular if and only if their dot product is zero.

As to "use" of dot product, the following seems plausible. Say we needed to determine the motion of a particle on a plane using Newtonian ideas such as forces. If we have a vector that lies in the plane and we know a vector perpendicular to the plane (how we know those is a different issue - and it's a whole lot easier to think of a 2D example) then we can resolve a force acting on a particle on the plane into two components, one along the plane and one perpendicular to the plane - which can be very useful in determining the motion of the particle if it remains in contact with the plane. See: here. I have a vague recollection that cross-product is useful when working with moving charges in electric or magnetic fields - there are probably many more.

Answer 4

In all fields, when people extend old terms into new contexts, the meaning often changes. That can be surprising in mathematics, because you want to feel like you can understand the basic concept and then extend it yourself. But it doesn't always work that way.

Simply put, you cannot multiply two vectors together in a way that is perfectly analogous to multiplying numbers. The dot product does not claim to be that kind of multiplication. When people take the dot product of two vectors, they say "I took the dot product", not "I multiplied them together". The dot product is its own thing. (Although, for 1-dimensional vectors it does work out to being the same as the regular product.)

And it's super useful. That's why they gave it a name. There are a lot of cases where a big vector coming at a large angle has a similar effect to a smaller vector coming in at a small angle, and the dot product handles those situations.

The cool and surprising thing is that $ab \cos \theta$ (if $a$ is the length of the first vector and $b$ is the length of the second, of course) works out to being the same number as when you multiply the two vectors component by component and add the results.