What is a dot product?

Question

When you have 2 vectors and you take the dot product, you end up with a scalar, but what does that scalar represent?
I have the formulas for the dot product, but can some one explain what the scalar represents?

So does that mean that when we project u onto d for example, we use the formula u dot d/ distance of d squared times d, so that we can get the vector for that projection instead of just a scalar?

Answer 1

The link below may give you a better visual representation of what a dot product represents.

Answer 2

Mathworld is great.

If you wanted something a little less formal, here's my 2 cents.

If you dot a vector V with a *unit* vector U, the number tells you how far V points in the direction of U.
If V points in the same direction, you get the length of V.
If V points in the opposite direction, you get the negative of the length of V.
If V is perpendicular (oops--I mean "orthogonal") to U, you get 0.

If U has a length other than one, the dot product scales accordingly. So, if U has length 2, then V dot U will be twice the distance V travels in the U direction.

You will probably need to look at several simple examples (mostly in 2-D) before this makes a lot of sense.

Enjoy!

-----

ZeroCarb, that visual was fly.

Answer 3

If you have a dot (or 'inner') product A∙B, then the scalar represents the projection of the length of A onto the direction of B. Think of it as the 'shadow' of A laying on B if the light source were shining directly 'down' on B. This is the reason the dot product of orthogonal vectors is 0.

HTH

Doug

Answer 4

Scalar Product of Vectors
The scalar product and the vector product are the two ways of multiplying vectors which see the most application in physics and astronomy. The scalar product of two vectors can be constructed by taking the component of one vector in the direction of the other and multiplying it times the magnitude of the other vector. This can be expressed in the form:


If the vectors are expressed in terms of unit vectors i, j, and k along the x, y, and z directions, the scalar product can also be expressed in the form:

Check on the source. I copied for you from there.
Good luck!

Answer 5

It can represent all kinds of things depending on what the original vectors are.

If vectors are force and distance then the dot product is work.

Note that the cross product of distance and force is a vector quantity torque!

Dot products are also the length of projections. This can show the length of your shadow given vectors representing your height and the normal vector to the ground's surface.

Answer 6

The dot product of two vectors, A=(A_1,A_2,...,A_n) and B=(B_1,B_2,...,B_n) is A.B=A_1B_1+A_2B_2+...+A_nB_n. When the dot product is zero, the vectors are perpendicular. From a geometry viewpoint, A.B=||A|| ||B|| cos(theta) where ||A|| is the length of vector A and theta is the angle between vectors A and B. This allows one to compute the angle between two vector from their coordinates, since
theta = arccos(A.B/||A|| ||B||).

Dot products have several properties, for instance:
(i) A.B = B.A (commutivity)
(ii) A.(B+C) = A.B+A.C (distributivity)
(iii) c(A.B) = (cA).B = A.(cB) (homogeneity)
(iv) A.A>0 for A\not=0 (positivity)
(v) A.A=0 iff A=0 (zero)
(vi) ||A|| = sqrt(A.A) (norm or length)
(vii) (A.B)^2 <= (A.A)(B.B) (Cauchy-Schwarz Inequality)
(viii) ||A+B|| <= ||A|| + ||B|| (Triangle Inequality)

The dot product of two vectors is a scalar (single real number).

Answer 7

http://mathworld.wolfram.com/DotProduct.html