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I have two vectors: $A = i-3j+2k$ and $B = -3i+4j-k.$

I have already found the angle between the two to be ~$157.6$ degrees, but I am required to find the smallest angle between the two - I am not sure what that means exactly and can't seem to find the answer online. I would be glad if you could help me out with that.

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They are position vectors in $\mathbb{R}^3$. Two vectors describe a plane so I presume you are being asked to find the angle between the two vectors in the plane they describe.

I've added a small illustration (done in Geogebra) to show what I mean. The two points $A$ and $B$ are described by the vectors you gave. The blue plane is described by $A, B$ and the origin and the angle between these two vectors is $153^{\circ}$ (not sure if the angle is properly visible). You can compute this angle via the dot product. If you used the dot product and got to $157.6^{\circ}$ there may be a rounding error of some description in your calculation. Either way, I believe this may be what your teacher is looking for. Does this help?

Plane

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    $\begingroup$ This would be more appropriate in the comments section. $\endgroup$ – MathematicianByMistake Mar 15 '16 at 20:22
  • $\begingroup$ I agree! My lack of reputation means that I am not yet allowed to comment everywhere. $\endgroup$ – Edward Evans Mar 15 '16 at 20:23
  • $\begingroup$ I am sorry, I did not remember that rule applies. For what it's worth, it's a usefull hint! $\endgroup$ – MathematicianByMistake Mar 15 '16 at 20:25
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    $\begingroup$ Thanks, I hope it helps the asker. Only 6 more reputation to go before I am deemed trustworthy enough to contribute to the community in the form of comments! Haha $\endgroup$ – Edward Evans Mar 15 '16 at 20:26
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    $\begingroup$ It's from my teacher's presentation. The answer is not indicated, sadly. $\endgroup$ – xen20 Mar 15 '16 at 20:48

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