Proving Vector Independence: A & B Not Parallel

dracolnyte
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Homework Statement


Show that two planar vectors a and b are linearly independent if and only if they are not parallel.

The Attempt at a Solution


I know that, if they are not parallel, they will meet and cross in a line.
What else should I know before proving this question?
 
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You certainly should know the definition of "independent vectors"! I don't think it will help to think "geometrically" here. Just use the definitions of "independent" and "dependent" vectors and the fact that, since the plane is two dimensional, two independent vectors must span the entire space.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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