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Having two vectors $a$ and $b$ and with $b$ fixed, how would I go about increasing the angle between the two vectors by a specified $\Delta \theta$?

The dot product equation:

$$ a \cdot b = |a| \cdot |b| \ \cos \theta .$$

gives multiple solutions in the form of a cone for $3$-dimensional vectors. I want the new vector $a'$ to be in the same plane spanned by $a$ and $b$.

Also can this case be extended to arbitrary number of dimensions?

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I feel like your question is a bit ambiguous. Increase the angle subject to a constraint? Without any constraint, as you mentioned, in $\mathbb R ^3$ this yields multiple solutions, just like in $\mathbb R ^2$ it would and certainly in higher dimensions as well. –  process91 Nov 9 '11 at 3:54
    
Ah, the constraint is that it must be in the plane spanned by $a$ and $b$. –  Chris D Nov 9 '11 at 4:06
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2 Answers 2

You can decompose $a$ into a part parallel to $b$ and a part perpendicular to $b$, then increase the perpendicular part. The parallel part of $a$ is $|a| \cos \theta \hat{b}=\frac{a\cdot b}{b^2}b$. The perpendicular part is the rest: $a-\frac{a\cdot b}{b^2}b$. $\tan \theta$ is the ratio of the magnitudes of these: $\frac{|a-\frac{a\cdot b}{b^2}b|}{|\frac{a\cdot b}{b^2}b|}$, so you can increase the perpendicular part enough to give the desired $\theta$

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Thanks, also if I wanted to preserve the length of the vector, how would I setup the ratio? At the moment, I can scale the perpendicular part to get the desired angle, but I need to preserve the original length. Essentially, I just wanted to rotate a vector away from another vector along the plane spanned by the two vectors. –  Chris D Nov 10 '11 at 15:35
    
Then you can just scale it back to the old length. If $a'$ is the new vector at the increased angle, $a''=a'\frac{|a|}{|a'|}$ will be in the new direction with the old length. –  Ross Millikan Nov 10 '11 at 16:06
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I thought more about your problem in general, without the restriction. and I think it is an interesting one. Here's what I came up with, intuitively.

Given two vectors, $a$ and $b$ in $\mathbb R ^n$, choose your coordinate axes to allow $b$ to be on one of the axes$^1$. Now spin $a$ around the axis which $b$ is on. This will create the cone you describe in $\mathbb R ^3$, and some other "surface" in other dimensions. Points on this surface have the same angle with $b$ as $a$. Vectors on the same side of the surface as $b$ have a lesser angle with $b$, and points on the side of the surface without $b$ have a greater angle with $b$.

I don't know if something interesting might come from algebraically trying this out in, say, $\mathbb R ^4$, but it should not be too difficult. Actually, in $\mathbb R ^3$, the cone of vectors with angle greater than that between $a$ and $b$ exists only if the angle between $a$ and $b$ is obtuse. If it is acute, you get the complement of a cone (all of $\mathbb R ^3$ minus the cone), and if the angle is $\frac \pi 2$, you get half of $\mathbb R ^3$.

If we take a two-dimensional "slice" of $\mathbb R ^3$, we would get either a triangle of infinite height, the complement of such a triangle, or a half-plane. In the higher dimensions, I would conjecture that a three-dimensional "slice" would always yield a cone, the complement of a cone, or a half-space, and a two-dimensional "slice" would therefore be similar to that already described.

$^1$ The purpose of this is just to make the algebraic manipulations easier. Obviously the choice of axes plays no direct role, and such a result is congruent to one in which $b$ is not placed on the axis.

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