Thanks for the great explanation. The mario kart analogy is superb.
I got confused by your analogy with projection or “shadow.”
Suppose vector A has coordinates (1, 1) and vector B has coordinates (2, 0). B is already on the X axis, so I don’t need to imagine rotating anything. The dot product of A and B is 12+10 = 2. But the projection analogy is telling me that the result should be 1 because vector A has an X coordinate of 1.
Hi Nolan, the tricky thing is realizing the size of the item being projected onto must be taken into account. If you like, imagine A and B as unit vectors, then scale them up.
i f*cking love you so much
It is also giving the sense that how to get DOT product… but if i want " how we can represent our DOT product in the vector space and what is the significance of that scalar quantity…? (some what related to the powers of signals and Euclidian scalars of length ) " is not explained . Hope you can explain and i am waiting for answer. Thank you.
Hi Bharadwaj, I’m not sure I understand the question. The dot product is a scalar quantity (just an number), but if you like, you can think of it as the size of the shadow of one vector on the other. (Remembering to “scale” the result by the magnitudes of the input vectors).
Yes kalid. 1st of all thanks for reply. And my perfect doubt is " shall i represent my dot product in the vector space that i multiplied the two vectors? " the answer may be possibly not because it is not having specific angle, but need some more justification. And in the most beautiful analogy of physics, i.e Fourier’s analogy, the 1st thing that we assumed was " signals are simillar to vectors " so , in that way i need some justifying point like, " why we took signals as similar to vectors? " and one more silly doubt… " what is the correct analogy between vectors and complex numbers? "
If any mistake in my thinking , sorry for inconvenience Kalid… (probably you are my banana brother… 
)
A.B=( A’s projection in the direction of B )×(B’s projection in the direction of A) is it right
Hey kalid ,… really i want explanation brother… when i m doing something , if the total process is just coming into my brain as a neat sketch… … it will be like your ahaa! moment right… please explain my doubts… ? if you think those are invalid … just give a reply like you can’t think like that … because of these reasons… please tell me bro… i m waiting for your explanation and my ahaa! moment… please…
On Nolan’s comment on June 23…and kalid’s response…" imagine A and B as unit vectors, then scale them up."
Could you elaborate?
Thx for the site. It’s been 20+ yrs since my last math or physics class and I miss the discussions.
@Brian: Thanks, glad the shadow analogy clicked!
@Jen: Sure thing. In math, it’s often easier to work out a fact for a simple system, then adjust as needed. Instead of “What’s 15% interest on $123,000?” you can ask “What’s 15% interest on $1?” (15 cents). Then you “scale it up” by multiplying by 123,000 – the scenario you care about. Sometimes it’s easier to work in small steps.
Similarly for vectors, imagine A and B as each having length of “1” (a single unit). Figure out the dot product between them, and then multiply it by the size of A, and then the size of B. At the end of the day you’ve accounted for everything, but it can be easier to think about unit vectors interacting, vs. vectors of arbitrary sizes.
This explanation of dot product is very clear and understandable !!!
From my point of view, the most interesting part is that of “Dot Product: Piece by piece” … this shows that THIS DOT PRODUCT IS THE ONLY “REASONABLE” PRODUCT BETWEEN TWO VECTORS THAT SATISFIES THE PROPERTY:
a perpendicular b if and only if a.b=0
under title dot product:rotate to base line
how vector(a) sin(theta) * vector(b) = zero
fallow figure and reply to my mail plz
Pretty cool article. I was looking for an interesting extension of vectors for my Honors Geometry class (we’ve already covered the basics) and they’ll appreciate the Mario Kart example when I introduce dot products.
Thanks, and I’ll be back for more reading soon!
Returning visitor, big fan, great stuff as always! One intuition for dot and cross products is that the dot product represents the “similarity of b to a”, whereas the cross product represents the “difference between b and a”, or what you would have to do to a to transform it to b, which includes that you have to turn it in a direction different from a. Similarity means the component that is parallel to a, where a and its direction are the “standard” by which b is measured.
A deeper intuition on these concepts is provided by Clifford Algebra, which treats all numbers as compound numbers, scalar plus vector, either of which can be zero. (same as all scalars can be considered as complex numbers with zero imaginary component). In Clifford Algebra the dot product is defined as (the average of) ab + ba, i.e. the sum of a number and its “reverse”, where (as in Quaternions) ab = -(ba), which cancels all directional components leaving only the scalar, whereas the “wedge product” (corresponding to the cross product) is defined as (the average of) ab - ba, subtracting the product from its reverse, where the reverse of the reverse is back to the original direction, which captures only the directional component, while the scalar component drops out. The dot product is the “symmetrical component” of the product, the wedge product is the “asymmetrical component”. THAT explains why the dot product is a scalar.
A lot of things become a lot more intuitive in Clifford Algebra.
a very good explanation for dot product ,banana ,apple example clear each point .we can also explain it by using W=f.scos¥ like part of force in the direction of distance moved by the object And work gives the amount of inc in distance covered due to force applied in that direction
thanks
One thing that confuses me about the dot product is the idea that, it is the amount of vector A applied to vector B. By itself, it’s not that bad, but the problem comes when I want to think of the reverse. Does it work the same way if I want to think how much of vector B is in vector A? if so, then what if vector B is bigger than A? is there a visual way to answer these questions
So this is what it was all about! Multiplying 2 vectors is like multiplying their components in the usual algebraic way: a=x1i + y1j and b=x2i + y2j. Then a dot b = (x1i + y1j) dot (x2i + y2j) = x1i dot x2i + x1i dot y2j + y1i dot x2i + y2j dot y1j. Since the unity vectors i and j are orthogonal, their dot-product will be 0, no matter what scalar they have. So we’re left with x1i dot x2i + y2j dot y1j. Neat! 
I like this response by Jim Smith, to a high school physics teachers question “What IS a dot product – what does it mean?”
To understand the dot product, I’d recommend learning about both the history of the ideas that led to its invention, and how the product is used in geometry. The first chapter of David Hestenes’s New Foundations for Classical Mechanics (http://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148/) is great for those purposes. His online primer on Geometric Algebra may also be useful, starting with the page http://geocalc.clas.asu.edu/GA_Primer/GA_Primer/introduction-to-geometric/defining-and-interpreting.html, where the “dot product” (referred to by the more-generic name “inner product”) is illustrated for the special case where one of the vectors is a unit vector. Dorst and Mann’s article “Geometric Algebra: A Computational Framework for Geometrical Applications” (https://staff.fnwi.uva.nl/l.dorst/cga-2.pdf) may also be useful.
Professor Hestenes is working to use Geometric Algebra (of which the dot product is an important element) to integrate high school algebra, geometry, trigonometry and physics into a coherent curriculum. Therefore, you won’t regret looking at the above references, even if they get a little deeper than you may need to at this time.
please make article about cross product as well,it’s really confusing no textbook explains cross product intuitively ,many books are only exam oriented .i hope u will definitely make article about cross product…thanks