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This question already has an answer here:

NOTE :

By perpendicular component of $\vec{F}$, I mean a vector which is a component of $\vec{F}$, but perpendicular to it.

enter image description here

In the image above, the red vectors are a possible set of rectangular components of $\vec{F}$, and blue vector is the sine component of cos component of $\vec{F}$, ie, perpendicular component of $\vec{F}$.

Now, logically, the perpendicular component of $\vec{F}$ should be zero, since its projection is zero.

But if we consider the image above, the Perpendicular component, $|\vec{F}|\cos\theta\sin\theta$, is not zero.

What is the cause of this discrepancy, and how to take components properly?

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marked as duplicate by David Z Mar 30 '16 at 17:00

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ The red arrows are just the components projected onto a pair of axes at an angle of $\theta$ to $F$. The real question is what the blue arrow is supposed to be... $\endgroup$ – lemon Mar 30 '16 at 16:51
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If you take again the components of the vector whose magnitude is $Fsin\theta$, you will see one component of it will cancel the vector represented by the blue arrow both of which will have a magnitude of $Fcos\theta sin\theta$:

enter image description here

The other components of the components of $\vec F$ will be vectors whose magnitudes are $Fcos^2\theta$ and $Fsin^2\theta$ whose sum will equal to $F$.

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