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How to decompose a vector into non orthogonal components?

I have a vector with coordinates (-4 , -1). I know how to decompose it into orthogonal components. But how can I decompose it into two vectors that have an angle of 75 degrees? How can I calculate the magnitude of the new two vectors? I don't need the graphical solution.

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1 Answer

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$(0,1)$ and $(\sin 75,\cos 75)$ are two vectors. Note that they are linearly independent, of norm one each and form a basis of $\mathbb{R}^2$ with angle $75^\circ$. To find the decomposition, we have to solve: $$ \begin{pmatrix} 0 \quad \sin 75 \\ 1 \quad \cos 75\end{pmatrix} \begin{pmatrix}x_1 \\ x_2\end{pmatrix} = \begin{pmatrix}-4 \\ -1\end{pmatrix} $$

Instantly, we get $x_2 = \dfrac{-4}{(\sin 75^\circ)}$, and then $x_1 = 4 \cot(75^\circ) - 1$. Hence, we can write: $$ (-4,-1) = (4 \cot(75^\circ) - 1)(0,1) + \dfrac{-4}{(\sin 75^\circ)}(\sin 75,\cos 75) $$

This is the decomposition you would have wanted. Remember, I could have taken any two vectors having an angle of 75 between them as a basis. Furthermore, there was nothing special about 75 above: you can replace 75 by 53 or 137 or 119 and get similar results.

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