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### Rotating Vectors

```
Date: 12/14/95 at 12:14:7
From: Anonymous
Subject: 3D Geometry? Projection?

I do not even know how to ask this question because I do not know the
names of things or concepts. I don't know where to start...

... but it is this:

If I were a vector, say [2,3,4], and I wanted to turn to face
another vector, say [-1,10,11], how would I do it? What would
I be doing ?

I hope you can help me because this REALLY bugs me...

Thanks in advance,
mlarch@fred.net
```

```
Date: 7/28/96 at 20:36:6
From: Doctor Jerry
Subject: Re: 3D Geometry? Projection?

Any two non-zero, non-collinear vectors issuing from the same point
form a plane. In this plane, there is an angle t between the two
vectors.  If we use the standard convention, 0 < t < pi.

It may be that you are asking how to rotate one vector so that it
falls along the same line as the other.  If so, I can tell you how to
calculate the angle between the two vectors you gave.  Use the law of
cosines on the triangle defined by the origin and the ends of the two
vectors.

The lengths of the vectors are sqrt(29) and sqrt(222).  The length of
the side opposite the angle t is sqrt(107).  So, from the law of
cosines,

107 = 222 + 29 - 2*sqrt(222)*sqrt(29)*cos(t).

Solving for cos(t),
cos(t) = 0.89733...
Hence, t = 0.457091... (radians) or 26.189409... (degrees).

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Linear Algebra

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