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I am learning Mechanics - motion in a plane.

Is it possible to that a given vector can be resolved in infinite ways into two non-colinear vectors in the same plane?

For example, I have a vector pointing in 190deg direction. Can I resolve it into two vectors - one pointing in 285 deg and another one pointing in 030deg direction.

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One way to prove it would be simply just do it the hard-way. Just find out the resolving vector. Lets assume there is a vector $\vec{C}$ which need to be resolved into vectors at direction $\alpha$ clockwise and $\beta$ counterclockwise to the Original vectors. Lets identify these directions by $\hat{A}_\alpha$ and $\hat{B}_\beta$ respectively.

$$ i.e.\ \ \vec{C} = A\ \hat{A}_\alpha + B\ \hat{B}_\beta $$

So all we need to do is to find out the values of $A$ and $B$ for any given $\alpha$, $\beta$-direction.

By taking dot product of the above equation with $\hat{A}_\alpha$ and $\hat{B}_\beta$ we get.

$$ C\ \cos\alpha = A + B\cos(\alpha+\beta) \\ and \\ C\ \cos\beta = A\cos(\alpha+\beta) + B $$

From the above equations it is easy to find out,

$$ A = \frac{\cos\alpha - \cos\beta\cos(\alpha+\beta)}{\sin^2(\alpha+\beta)} C \\ and \\ B = \frac{\cos\beta - \cos\alpha\cos(\alpha+\beta)}{\sin^2(\alpha+\beta)} C $$

Now just keep plugging the numbers of $\alpha$ and $\beta$ to get $A$'s and $B$'s. Since there are infinite choices of $\alpha$ and $\beta$ hence there are infinite ways to resolve a vector.

NOTE: Even though a vector can be resolved in infinite ways. For given set of the directions (i.e. given $\alpha$ and $\beta$) the resolution is unique and unambiguous.

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