Resolving vectors into components

The problem is to determine the components of $F_2$.

Method 2

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The components with respect to horizontal and vertical axes? – rschwieb Feb 5 '13 at 19:48
Along the u, v axes. – miles Feb 5 '13 at 19:49

Agreed, @5PM.

Here's a general approach to vector problems that I've found useful, using the magnitude-angle notation.

There is the original vector, 150$\angle$165°, with x-component $150 *$cos$(165°)$and y-component $150 *$sin$(165°)$

This is to be broken down into two non-orthogonal "components", F$_v$$\angle90°, and F_u$$\angle$195°

Each of these two last can be separately broken down into x and y components in the same way as the 150 N force, above.

Then the 150 N x-component equals the $F_v$ x-component plus the $F_u$ x-component, and the 150 N y-component equals the $F_v$ y-component plus the $F_u$ y-component. Two equations, two unknowns, all angles known , so just solve. Oh, and one equation has only one unknown!

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That's a nice way of thinking about components. Doing so allows me to calculate ("wolframalpha.com/input/?i=Solve[{uCos[195]%2BvCos[90]%3D%3D150*Co‌​s[165]%2CuSin[195]%2BvSin[90]%3D%3D150*Sin[165]}%2C{u%2Cv}") – miles Feb 6 '13 at 8:54

Method 1 is correct. In method 2 you calculated orthogonal projections of the force $F_2$ onto the given axes. However, the components of a force coincide with orthogonal projections only when the axes are mutually orthogonal.

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True; that's why I think the original question is worded in a confusing manner, in that it refers to the two, non-orthogonal desired vectors as "components" of the original vector.(See the first comment on the question by rshwieb, above, and the reply by the OP referring to the u and v directions as axes) The method I proposed treated all three vectors the same, resolving each one into two orthogonal, x-y components that could then be dealt with as scalars. – DJohnM Feb 6 '13 at 16:05
@User58220 From mathematical point of view, there is nothing wrong in calling non-orthogonal vectors components. The vectors $u,v$ form a basis of $\mathbb R^2$; they are distinguished by this designation. Given any other vector $w$, we can uniquely decompose/expand it as $w=au+bv$. It is natural to refer to $au$ and $bv$ as components of $w$ in this basis. What else should we call them? // Note that the existence of such expansions has nothing to do with orthogonality: it makes sense in vector spaces that have no inner product, and therefore no concept of orthogonality. – user53153 Feb 6 '13 at 16:59