Whats difference between inner product and dot product?

[aditya23456]

i m really confused..please explain with a physical example so that I can learn the application of it

 

[Tinyboss]

The dot product is a particular example of an inner product. Or, equivalently, the notion of inner product generalizes the dot product.

A physical example is that in Euclidean space, the dot product of two vectors is equal to the cosine of the angle between them.

 

[DonAntonio]

The dot product is a particular example of an inner product. Or, equivalently, the notion of inner product generalizes the dot product.

A physical example is that in Euclidean space, the dot product of two vectors is equal to the cosine of the angle between them.

...times the product of their lengths !

DonAntonio

 

[Rasalhague]

If you're fairly new to vectors, and have encountered them so far only in the context of elementary physics, as things "with a direction and magnitude", as arrows that can be added together with the parallelogram rule, or as a set of components

$$\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{i}$$

or as a column matrix

$$a = \begin{pmatrix}a_1\\ a_2\\ a_3\end{pmatrix}$$

then you can regard inner product as just another name for dot product. In this context, scalar product also means the same thing. A physical example is the work done by a constant force $\mathbf{F}$ on an object displaced along the vector $\mathbf{d}$:

$$W = \mathbf{F} \cdot \mathbf{d}.$$

If you go on to study more advanced math and physics, you'll encounter other types of vector, and other types of inner product. Till then, don't worry about it. But if you're curious...

Given a vector space V over a field K (where K may be the real numbers R, or the complex numbers C), an inner product is any map g:VxV-->K that obeys certain axioms.

We can talk about "the inner product of a pair of vectors" when the vectors belong to an inner product space; that is, a vector space for which a particular inner product has been chosen.

The vector space Rⁿ, consisting of all ordered lists of n real numbers, (x₁,x₂,...,x_n), with componentwise addition, is usually made into an inner product space with the map g:RⁿxRⁿ-->R such that

$$g(a,b) = \sum_{i=1}^n a_i b_i.$$

This inner product is often called the dot product. So in this context, inner product and dot product mean the same thing. But inner product is a more general term than dot product, and may refer to other maps in other contexts, so long as they obey the inner product axioms.

Vectors in Rⁿ also be viewed as directed line segments (arrows) from the origin. Viewed in this way, the dot product can be defined by the following rule of assignment:

$$g(a,b) = ||a|| \cdot ||b|| cos(\theta)$$

where ||x|| is the norm of the vector x, and $\theta$ the angle between vectors a and b. This definition is equivalent to the other; that is, it always gives the same result.

 

[HallsofIvy]

The theoretical "meat" of the Gram-Schmidt orthogonalization process is: given any inner product, there exist a basis such that the inner product is the same as the "dot product" of coefficients in that basis.