Thanks Kalid! That really helped me get my head around the concept!
Hi Kalid,
Your site is excellent, thank you for your resources. I am a high school math teacher, and I am using this article to motivate how I want to now teach the dot product.
My question is: How can we distinguish a situation that would call for vector ADDITION between one that would call for using the dot product. For example, the classic airplane flying in one direction and the wind blowing in the other requires us to add the wind vector to the airplane vector and find the magnitude and direction angle of the vector of the sum. Why wouldn’t the dot product be appropriate when analyzing such a situation, and why does the Mario Kart boost situation not call for vector addition but rather the dot product? Are both appropriate, but just tell us different things about the situation?
Thanks in advance!
Brendon
Hi Brendon, thanks for the comment, I love helping out other teachers.
It basically comes down to whether regular addition or multiplication would work for the scenario. Imagine everything is moving in the same direction (along the x-axis), so we can just use regular numbers. Would we add or multiply?
In the airplane scenario, we’d add the speed (500mph) to the wind speed (-50mph). In Mario Kart, I’m assuming we multiply the effect of the boost pad (2x) with your incoming speed (20mph). So, if you’re simply placed on the boost pad (0 speed) you don’t see any change. Only when you’re driving do you notice the impact.
(This probably isn’t how it actually works, the pad may add some speed no matter how fast you’re going, or maybe it accelerates everyone to a certain speed unreachable in the normal game.)
The dot product is basically “multiplication, taking direction into account”. So I like to simply the scenario to regular numbers and see if multiplication or addition is needed.
you are a king man, can u do something like this on combining matrices and vectors or how the vector matrix stuff works?
you are the best man, and I am not trying to boost ur ego, seriously the ability to reduce abstract stuff into oranges and bananas is a gift.
Hi Kalid,
I have just finished reading your book and I have now found your website. It is a Godsend. I wish I had known about it back when I was in grad school. My whole life might have taken a different path.
I am also in the matrix algebra class offered by edx. Both the course and your site are helpful but only your site is actually fun.
I noted in the comment you made back in 2012 that you didn’t have a good intuitive explanation for eigen values and eigen vectors. If anything has occurred to you I would be more than grateful. And the determinant! What is that!?
It is like you are the Catcher in the Rye, saving lost children before they run off the cliff. Seriously, this site is not only useful, it makes me happy!
It still doesn’t Click… Why IS the AxBx + AyBy actually equal to the projection of A on B times B? It looks like the result is still a vector of x and y (or apples and oranges) but the endresult is a scalar… how come?
@Imre: Great question. We can write any vector as the sum of its projections:
A = (Ax, 0) + (0, Ay) [A is the sum of it’s x-axis projection and y-axis projection]
B = (Bx, 0) + (0, By)
We can take the dot product between these forms:
[(Ax, 0) + (0, Ay)] dot [(Bx, 0) + (0, By)]
which reduces to:
Ax.Bx + Ay.By
since the interaction between different dimensions is zero. The dot product is defined (i.e., we just decide) to track Ax.Bx as a scalar quantity just measuring the overlap, and Ay.By as another scalar quantity. With the cross product, we decide that the interactions stay as vectors, and the result of the cross product is a vector.
It’s possible to have a dot product variant that keeps Ax.Bx and Ay.By separate (not sure if there’s a name for it) but the dot product, as traditionally defined, decides to use scalars.
excuse me . Mr \ khalid . I want 10 ways to calculate dot product ? . because my math doctor in faculty asked that .