A type of vector multiplication. (see also cross product) Usually given the symbol "." (hence the name). Defined as: A . B = |A| |B| cos(t) where A and B are vectors and t is the angle between them. Notice that unlike it's brother the cross product the result of this multiplication is a scalar.

The dot product (denoted with *) of n-length vectors a and b defined as (a1,a2,...,an)*(b1,b2,...,bn) = a1b1+a2b2+...+anbn. It is the vector equivalent of scalar multiplication; its result happens to be equal to ||a||*||b||cos(t) where t is the angle between the vectors. ||v|| denotes the magnitude of v; note that ||v|| = sqrt(v*v) much as how |x| = sqrt(x*x).

You can think of a linear transformation as just a bunch of dot products. For example, here is a matrix describing a linear transformation that rotates vectors on a two-dimensional plane:

```(cos(phi)  sin(phi))
(-sin(phi) cos(phi))
```
When you multiply a two-dimensional column vector with this matrix, you take two dot-products, one with the upper row and one with the lower row. The first row of this matrix is an unit vector, which makes angle phi with the x-axis. The second row is orthogonal to the first row, which is easy to check by taking dot products. So if you think about it a bit, you see that this transformation must be either a rotation or a combined rotation and 'mirroring' of the vecto r around either axis. If you do some more thinking you can convince yourself that this particular transformation does not 'mirror' the vector, unlike this matrix, which mirrors a vector with respect to the x-axis:
```(-1 0)
(0  1)
```

It should be noted that, even though the definitions of the dot product given above are correct, the angle, theta, is the angle between the two vectors when they are attached initial point to initial point.

Also, a vector noted with ||'s on it means the magnitude or absolute value of that vector. It can be found using the length formula of: |a| = sqroot(a1^2+a2^2+...+aN^2)

In sum,

a.b = |a||b|cos(t), or a.b = a1b1 + a2b2 + ... + anbn

where a and b are two vectors in R^n and t is the angle enclosed when the two vectors are attached initial point to initial point.

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