- #1

etotheipi

As far as I know, a set of vectors forms a basis so long as a linear combination of them can span the entire space. In ##\mathbb{R}^{2}##, for instance, it's common to use an orthogonal basis of the ##\hat{x}## and ##\hat{y}## unit vectors. However, suppose I were to set up a basis (again in ##\mathbb{R}^{2}##) with two vectors which weren't orthogonal, ##\hat{a}## and ##\hat{b}##. The vectors can still span the space, so everything seems fine. However, if I try to express a vector law in terms of this basis, like

$$\vec{F} = m\vec{a} \implies F_{a}\hat{a} + F_{b}\hat{b} = ma_{a}\hat{a} + ma_{b}\hat{b}$$

then a few problems arise. If we compare the coefficients of any given component, the result is valid (i.e. ##F_{a} = ma_{a}##). However the overall expression is just wrong.

For one, the magnitude of ##\vec{a}## is not ##\sqrt{a_{a}^2 + a_{b}^{2}}##; we know this because to compute the real magnitude we'd need to use the cosine rule. However, I thought that a property of vectors was that the magnitude is the root of the sum of the squares of the components (although I'm assuming this only applies if the basis is orthogonal...).

So then I'm conflicted, since it seems valid to use a basis with basis vectors that are not orthogonal, but now the usual rules of vectors don't apply (or at least they need to be tweaked slightly). So is that to say we can only use an orthogonal basis? Thank you!

$$\vec{F} = m\vec{a} \implies F_{a}\hat{a} + F_{b}\hat{b} = ma_{a}\hat{a} + ma_{b}\hat{b}$$

then a few problems arise. If we compare the coefficients of any given component, the result is valid (i.e. ##F_{a} = ma_{a}##). However the overall expression is just wrong.

For one, the magnitude of ##\vec{a}## is not ##\sqrt{a_{a}^2 + a_{b}^{2}}##; we know this because to compute the real magnitude we'd need to use the cosine rule. However, I thought that a property of vectors was that the magnitude is the root of the sum of the squares of the components (although I'm assuming this only applies if the basis is orthogonal...).

So then I'm conflicted, since it seems valid to use a basis with basis vectors that are not orthogonal, but now the usual rules of vectors don't apply (or at least they need to be tweaked slightly). So is that to say we can only use an orthogonal basis? Thank you!

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