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Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to rotate the vector from local space to world space?

For ease of use, the values are:

Vector v = $[1.0, 0.0, 0.0]$

Quaternion $q = [W: 0.7071068, X: 0, Y: 0.7071068, Z: 0]$, which I understand to be a rotation $90^\circ (\frac{\pi}{2})$ around the $Y$-axis and which converts from the local space to the world space. (That is, the resulting vector is $[0.0, 0.0, 1.0]$, and if this was the nose of a spaceship, it'd be pointing to the right in world coordinates)


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@Narf: In response to your previous question, I'd linked to this Wikipedia article in my answer: Could you say more specifically what part of the explanation there you don't understand? – joriki May 19 '11 at 22:45
I'm not rotating the quaternion; I'm rotating the vector. Seems like it'd be a different thing. – Narf the Mouse May 20 '11 at 15:02
@Narf: No, that's exactly what's described there; I'd even linked to the relevant section of the article in my answer to your previous question:… – joriki May 20 '11 at 22:46
@joriki, how is v' = qv(q^-1) represented in terms of the x, y, z coordinates in the vector v in relation to the w, x, y, z coordinates in the Quaternion for a rotation of v by q. – dcousens Jun 3 '11 at 12:01
@Daniel: If I understand your question correctly, again, the answer is right there in the Wikipedia article I twice linked to: "Let also $\vec{v}$ be an ordinary vector in 3-dimensional space, considered as a quaternion with a real coordinate equal to zero." – joriki Jun 4 '11 at 13:09
up vote 17 down vote accepted

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:

and this:

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.

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Again, I'm not asking how to rotate a quaternion by a vector this time; I'm asking how to rotate a vector by a quaternion, resulting in a vector pointing in a different direction. – Narf the Mouse May 20 '11 at 15:04
Since the "vector" in your question has three coordinates, just put a 0 in front to get a pure quaternion. – Will Jagy May 21 '11 at 17:59
Of course he knew what the OP was asking. The OP was the only one having trouble understanding that he was being given answers that were valid; he was getting hung up on terminology, and Will Jagy merely pointed that out and offered resources to try to clear up that confusion. – Muphrid Oct 22 '13 at 2:46
'I have no idea what engineers and programmers call these concepts. You are asking mathematicians.' Yes, he does. Rotations in 3d space. Being 'we're mathematicians we don't deal with real world things' is unhelpful to new comers. – Doug Nov 24 '13 at 6:29
Thanks; this gave me some mappings between concepts — scare quotes and all — that made the reference Wikipedia material a lot easier to understand. – Yuki Izumi May 5 '14 at 11:20

The simple answer to the question is, given:

 R = [w, x, y, z]  
 P = [0, p1, p2, p2]
 R' = [w, -x, -y, -z] 

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

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

In the example given these are:

 R = [0.7071203316249954, 0.0, 0.7071203316249954, 0.0]
 R' = [0.7071203316249954, 0.0, -0.7071203316249954, 0.0]
 P = [0, 1, 0, 0]
 H(R, P) = [0.0, 0.7071203316249954, 0.0, -0.7071203316249954]
 H(H(R, P), R') = [0.0, 0.0, 0.0, -1.0000383267948871]

Notice the result is a length 4 vector; the w component will always be zero and can be discarded.

Ie. The point (1, 0, 0) is transformed by a rotation around the Y-axis by 90 degrees and becomes (0, 0, -1)

For those unfamiliar with quaternions, it's worth noting that the quaternion R is generated as described here (

a = angle to rotate
[x, y, z] = axis to rotate around

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

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],

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])
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