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I have two vectors $a\hat j$ and $b\hat i$ and the plane $x+y+z=1$. I want to find the components of the vectors perpendicular to the plane.

Now as far as I know, the unit normal vector to the plane is $\hat n=\frac{1}{\sqrt3}(1,1,1)$. So, the perpendicular component of the first vector should be $(\bar a \cdot\hat n)\bar a$ which is $\frac{a^2}{\sqrt3}\hat j$. Same goes for the second vector.

But is this correct? The direction is the same as before. What am I doing wrong?

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The direction is the same as before because you calculated a multiple of the original vector instead of a multiple of the unit vector. You want $(\bar a\cdot\hat n)\hat n$ instead of $(\bar a\cdot\hat n)\bar a$.

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  • $\begingroup$ Oh..! So what I calculated is actually the component of the normal along the first vector. Got it! $\endgroup$ Commented Apr 17, 2016 at 10:17
  • $\begingroup$ @Tejas: Only if the first vector is normalised. Otherwise it's the component of the normal along the first vector, multiplied by the square magnitude of the first vector. $\endgroup$ Commented Apr 17, 2016 at 10:27

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