What is Dot product: Definition and 388 Discussions

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called "the" inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle of two vectors is the quotient of their dot product by the product of their lengths).
The name "dot product" is derived from the centered dot " · ", that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector, as is the case for the vector product in three-dimensional space.

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1. Help me understand why this summation index is not j

The below image is an excerpt from a website about Markov Chains. In the red boxed which I put in the image, I don't understand why the term ##g(i)## isn't being summed over ##j## instead of ##i##, since the outer sum is over the ##i##th element of the vector ##Pg##, which is the dot product...
2. B Confused about dot product multiplication

I'm confused about what we are really measuring when taking the dot product of two vectors. When we say we are measure "how much one vector points in the direction of the other", that description is not clear. At first I thought it meant how much of a shadow one vector casts on another and I...
3. Given value of vectors a,b, b.c and a+(b×c), Find (c.a)

I thought this was too easy $$a+(b\times c)=0\implies a=-(b\times c)=(c\times b)$$ Then $$3(c.a)=3(c.(c\times b))=0$$ Since cross product of vectors is perpendicular to both vectors and dot product of perpendicular vectors is zero. Now here's the problem, correct answer given is 10. But how do...
4. I Inner product vs dot/scalar product

Hi, from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things. Given a vector space ##V## an inner product ## \langle . | . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently...
5. I Dot product, inner product, and projections

In simple Euclidean space: From trig, we have , for u and v separated by angle Θ, the length of the projection of u onto v is |u|cosΘ; then from one definition of the dot product Θ=arcos(|u|⋅|v|/(u⋅v)); putting them together, I get the length of the projection of u onto v is u⋅v/|v|. Then I...

48. Magnetic moment and magnetic field and dot product

Homework Statement a magnetic moment of U = 1 (i) + 2 (k) , surrounded by a magnetic uniform field of B= 3 (i) + 4 (j) - 1 (k) find the potential energy in mJ ? Homework Equations dot product of 2 vectors ( ui*bi)+(uj*bj)+(uk*bk) = or finding the module of both vector and doing AB...
49. B Find Perpendicular Forces Given 20 N at 60 Degrees

I have a question. If you're given a force of 20 N and its 60 degrees to the horizontal, how could you find two perpendicular forces?
50. I Differentiating vector dot product

Hi. If I have a vector v , say for velocity for example then v.v = v2 and I differentiate wrt t v.v I get 2v.dv/dt but if I differentiate v2 I get 2v dv/dt but v.dv.dt is not the same as v dv/dt so what am I doing wrong ? Thanks