# What is the relationship between the axes of the coordinate system used for vector resolution?

This question is with regards to the statement made around page 56 (1971 Edition) in Anthony French's Newtonian Mechanics. He is discussing the choice of a coordinate system where the axes are not necessarily perpendicular to each other. Here is the summary of what I read (as applied to vectors in a two-dimensional plane):

A vector $$\vec{A}$$ makes the angles $$\alpha$$ and $$\beta$$ with two coordinate axes (we'll still call the axes $$x-$$axis and $$y-$$axis respectively) not necessarily perpendicular to each other (i.e. $$\alpha+\beta$$ is not necessarily $$\pi/2$$). Then, $$A_x = A\cdot\cos\alpha$$ and $$A_y=A\cdot\cos\beta$$, where $$A_x, A_y$$ are the magnitudes of the $$x, y$$-components of $$\vec{A}$$ respectively. In the generalized two-dimensional case, we have the relationship $$\cos^2\alpha+\cos^2\beta = 1$$.

I have redrawn the accompanying figure: How does French arrive at the generalized relationship: $$\cos^2\alpha+\cos^2\beta = 1$$ in two dimensions and $$\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1$$ in three dimensions?

In the case of perpendicular axes in two dimensions, it is clear why it would hold, since $$\alpha+\beta=\frac{\pi}{2}$$. But I am not sure how it holds in general.

• Maybe the anonymous downvoter intends to say that I am misreading the text? I do believe, however, that French should have been clearer. – Kedar Mhaswade Mar 31 at 4:27
• Must $x-$ and $y-$ axes be perpendicular always? – Kedar Mhaswade Mar 31 at 4:28