Lisa Hardin

2022-07-20

I want to find the vertical or you could say perpendicular component of $\stackrel{\to }{a}$ on $\stackrel{\to }{b}$
Now I know that it can be found out using $\stackrel{\to }{a}-\left(\frac{a\cdot b}{|b|}\right)\stackrel{\to }{b}$
However I wanted to know why it cannot be found out using this method I tried. What is the flaw in it?
$\stackrel{\to }{a}×\stackrel{\to }{b}=|a||b|\mathrm{sin}\theta \stackrel{^}{n}$ Now $\stackrel{^}{n}$ should be equal to $\frac{\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)}{|\stackrel{\to }{a}×\stackrel{\to }{b}|}$
Using this I can write the expression as $\stackrel{\to }{a}×\stackrel{\to }{b}=|a||b|\mathrm{sin}\theta ×$ $\frac{\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)}{|\stackrel{\to }{a}×\stackrel{\to }{b}|}$ which on simplifying gives me
$\frac{|\stackrel{\to }{a}×\stackrel{\to }{b}|}{|b|}=|a|\mathrm{sin}\theta$ which I believe should be the perpendicular component
Now I'm probably doing something really stupid but I can't really understand where am I going wrong ?

fairymischiefv9

Expert

You are correct that the maginitude of the transverse component of $\stackrel{\to }{a}$ relative to the direction $\stackrel{^}{b}=\stackrel{\to }{b}/|\stackrel{\to }{b}|$ is given by $|\stackrel{\to }{a}×\stackrel{^}{b}|$ for ${\mathbb{R}}^{3}$. However, this does not tell you the direction of of the transverse component.
It's direction is not along $\stackrel{^}{n}$, $\stackrel{\to }{a}\cdot \stackrel{^}{n}=0$; it's direction is along $\stackrel{^}{n}×\stackrel{^}{b}$
Of course, all of this cross-product stuff only works in 3 dimensions.

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