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I need help finding the unit vectors. How would I:

List all the unit vectors that are linearly dependent with: $[5;12]$

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Do you know what "unit vector" means? Do you know what "linearly dependent" means? –  Chris Eagle Mar 2 '13 at 21:55
    
I understand unit vector-> sqrt(x^2 + y^2) = 1. Not sure how to set up these equations. What does linearly dependent give us? –  Allen Miller Mar 2 '13 at 21:57
    
Again: do you know what "linearly dependent" means? –  Chris Eagle Mar 2 '13 at 21:57

3 Answers 3

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Strong hint: You have the vector $$ v = \pmatrix{5\\ 12} $$ given and you want to find all unit vectors (i.e. vectors of length $1$) that are parallel with $v$ (two vectors are linearly dependent exactly when they are parallel). So you want $$ w = \pmatrix{x \\ y} $$ such that $\lvert w\rvert^2 = 1$ and $v\cdot w = \lvert v\rvert\lvert w\rvert\cos(0\text{ or } \pi)$, so that means $$\begin{align} x^2 + y^2 &= 1 \quad\quad\text{and}\\ v\cdot w = 5x + 12y &= \pm\lvert v\rvert = \pm\sqrt{5^2 + 12^2} = \pm\sqrt{169}. \end{align} $$ That is: $$\begin{align} x^2 + y^2 &= 1\quad\quad\text{and}\\ 5x + 12y &= \pm 13. \end{align} $$ Now try to solve those two equations for $x$ and $y$.

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I'm not sure how you got these equations? Since we want a unit vector we can use norm([x;y])= 1 , so sqrt(x^2 + y^2)= 1. I then used s* [x;y] = [5;12]. Not sure how you got 13? –  Allen Miller Mar 2 '13 at 22:37
    
@AllenMiller: Sorry for the slow response. I updated my answer a bit. The $x^2 + y^2 = 1$ comes from the fact that $\lvert w \rvert = 1$. So that means that $\lvert w \rvert^2 = 1$. The $13$ comes from the $\sqrt{169}$. I had by the way made a mistake in the earlier answer. I bad forgotten that the dot product $w\cdot v$ can be plus or minus $\lvert v\rvert$. Does it make sense now? –  Thomas Mar 2 '13 at 23:25

The question is clear. there are two unit vectors (vectors of length 1) which are linearly dependent (are linear combination of one vector - this means, both solutions are only obtained from the given vector by multiplication)

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Alternatively, suppose you have two unit vectors in the direction of (5,12), show that these are equal up to sign. Therefore there are only two unit vectors parallel to (5,12) and you can find these by dividing (5,12) by its norm.

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the vector of the opposite direction is also the answer to the main question. –  V-X Mar 2 '13 at 22:02
    
thank you added to my answer. –  newToProgramming Mar 2 '13 at 22:03

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