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# Rotate a Vector3 direction

How can i rotate a Vector3 direction 45 degrees along the y axis? So a Vector3(1,0,0) for example, would become Vector3(0.5,0,0.5).

And i cant use Transform.RotateAround, or any other Transform methos. As i am not applying this Vector3 directly to a object.

**Answer** by Jessy
·
Feb 08, 2011 at 08:04 PM

http://unity3d.com/support/documentation/ScriptReference/Quaternion.AngleAxis.html

```
vector = Quaternion.AngleAxis(-45, Vector3.up) * vector;
```

That's a general case. For rotation around a world axis, it's faster/easier:

```
vector = Quaternion.Euler(0, -45, 0) * vector;
```

For (1,0,0), this results in (sqrt(.5), 0, sqrt(.5)), not (.5,0,.5), by the way. The length of the rotated vector stays constant. To achieve a squared hypotenuse of 1, you add .5, 0, and .5, not .25, 0, and .25, which would shorten your vector.

Glad to hear it. Rotate those vectors like they were on the Witches' Wheel!

Thanks! Worked great.

One thing to note is that you cannot do:

```
vector = vector * Quaternion.AngleAxis(-45, Vector3.up);
```

and must do

```
vector = Quaternion.AngleAxis(-45, Vector3.up) * vector;
```

as there are some reasons internal to Unity.

as there are some reasons internal to Unity.

Nope, reasons internal to mathematics. You wouldn't say that 2/3 and 3/2 are different "because Unity", would you?

That is because Quaternion Multiplication, A x B is not B x A. This is just the tip of the iceberg when you talk about complexity and beauty of quaternions.

Right, though Unity just decides to only implement Quaternion * Vector multiplication. Mathematically in order to rotate a vector you have to multiply the quaternion on the left side of the vector and the complex conjugate on the right side which is all implicit in that one multiplication operator in Unity. To see an actual run-down I always recommend this Numberphile video. It does simplify the concept a lot, but you get a clue what's actually going on. I once manually derived the quaternion - vector multiplication myself by literally expanding and then simplify the following

```
(q.w + q.x*i + q.y*j + q.z*k) * (v.x*i + v.y*j + v.z*k) * (q.w - q.x*i - q.y*j - q.z*k)
```

though always remember the rules i² = j² = k³ = ijk = -1 and the order. So `i*j == k`

but `j*i == -k`

. i,j and k form a "ring". Writing them in a circle clockwise and you can derive the sign easily.

```
// clockwise
i*j = k
j*k = i
k*i = j
// counter clockwise
k*j = -i
j*i = -k
i*k = -j
```

in the end the real parts should cancel and you're left with just i,j and k which is your rotated vector.

Here's my run-down. The resulting equations essentially form a 3x3 matrix.

**Answer** by elhispano
·
Feb 10, 2013 at 03:00 PM

That operation is dependent on the order.
rotatedVector = Quaternion * vector OK
rotatedVector = vector * Quaternion won´t works and will launch a compiler error

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