How do you rotate a vector by a unit quaternion?

To answer the question simply, given:

 P  = [0, p1, p2, p3]  <-- point vector
 R  = [w,  x,  y,  z]  <-- rotation
 R' = [w, -x, -y, -z]

For the example in the question, these are:

 P  = [0, 1, 0, 0]
 R  = [0.707, 0.0,  0.707, 0.0]
 R' = [0.707, 0.0, -0.707, 0.0]

You can calculate the resulting vector using the Hamilton product H(a, b) by:

 P' = RPR'
 P' = H(H(R, P), R')

Performing the calculations:

        H(R, P)      = [0.0, 0.707, 0.0, -0.707]
 P' = H(H(R, P), R') = [0.0, 0.0,   0.0, -1.0  ]

Thus, the example above illustrates a rotation of 90 degrees about the y-axis for the point (1, 0, 0). The result is (0, 0, -1). (Note that the first element of P' will always be 0 and can therefore be discarded.)

For those unfamiliar with quaternions, it's worth noting that the quaternion R may be determined using the formula:

a = angle to rotate
[x, y, z] = axis to rotate around (unit vector)

R = [cos(a/2), sin(a/2)*x, sin(a/2)*y, sin(a/2)*z]

See here for further reference.

You seem to be having a good deal of trouble with this, over several questions. At the same time, I am confident that you will get no satisfying answers as long as you stick with the terminology you are using. Maybe on the original stack overflow site, aimed at programmers.

Please read this: http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

and this: http://en.wikipedia.org/wiki/Quaternions

To answer not much more than your question, any quaternion is an expression $$ q = w + x \; \mathbf{i} + y \; \mathbf{j} + z \; \mathbf{k}.$$ where the multiplication rules use $$ \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i} \mathbf{j}\mathbf{k} = -1, $$ and consequences of those, see wikipedia as I said. Any quaternion $ q = w + x \; \mathbf{i} + y \; \mathbf{j} + z \; \mathbf{k}$ has a conjugate, that on wikipedia is written $q^\ast,$ given by $$ q^\ast = w - x \; \mathbf{i} - y \; \mathbf{j} - z \; \mathbf{k}.$$

The "norm" of the quaternion $q$ is exactly $$ \parallel q \parallel^2 = w^2 + x^2 + y^2 + z^2 = q q^\ast = q^\ast q$$

A quaternion $q$ is called a "unit" quaternion when $$ w^2 + x^2 + y^2 + z^2 = 1. $$

A quaternion is called "pure" or a vector in 3-space when $ w = 0,$ so a vector in 3-space is $$ v = v_1 \; \mathbf{i} + v_2 \; \mathbf{j} + v_3 \; \mathbf{k} $$ I have no idea what engineers and programmers call these concepts. You are asking mathematicians.

Given two quaternions, the norm of the product is the product of the norms.

The "real part" (the $w$) of the product of two quaternions $pq$ is the same as the "real part of $qp.$

So, what happens when I take a unit quaternion $q$ and a "pure" quaternion $v,$ and calculate $$ p = q^\ast v q.$$

Well, we have $$\parallel p \parallel = 1 \cdot \parallel v \parallel \cdot 1 = \parallel v \parallel $$

But as to the "real part," we begin with $$ \Re v = 0,$$

then $$ \Re q^\ast (v q) = \Re (v q) q^\ast = \Re v (q q^\ast) = \Re v = 0. $$

So $ p = q^\ast v q$ is another pure quaternion, another "vector," the same length as $v,$ but rotated from where it was.

That's enough for a start.

The answer above from Doug has been extraordinarily helpful to me. What is difficult with many explanations of Quaternions and their applications is that the notation is difficult to read and understand for learners.

This site from Mathworks explains further how to multiply Quaternions: http://www.mathworks.com/help/aerotbx/ug/quatmultiply.html?requestedDomain=www.mathworks.com

I also tend to think and learn in terms of code. Here is what I was ultimately able to piece together in python:

def quaternion_mult(q,r):
    return [r[0]*q[0]-r[1]*q[1]-r[2]*q[2]-r[3]*q[3],
            r[0]*q[1]+r[1]*q[0]-r[2]*q[3]+r[3]*q[2],
            r[0]*q[2]+r[1]*q[3]+r[2]*q[0]-r[3]*q[1],
            r[0]*q[3]-r[1]*q[2]+r[2]*q[1]+r[3]*q[0]]

def point_rotation_by_quaternion(point,q):
    r = [0]+point
    q_conj = [q[0],-1*q[1],-1*q[2],-1*q[3]]
    return quaternion_mult(quaternion_mult(q,r),q_conj)[1:]

print(point_rotation_by_quaternion([1, 0, 0],[0.7071203316249954, 0.0, 0.7071203316249954, 0.0])