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I only just now realized that you said perpendicular rather than parallel. A plane cannot be perpendicular to two lines going in different directions. This problem only makes sense if the plane is parallel to the two given lines. Then the normal vector is $[1,2,-1]\times[2,1,-3]=[-5,1,-3]$ and the equation of the plane is $-5(x+1)+(y-4)-3(z+2)=0$ ...


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Since the plane is perpendicular to the given lines, then the direction vector of these lines are normal to the plane. Take, for instance, the vector $\vec{N} = (1,2,-1)$. We know the point $P(-1,4,-2)$ lies in the plane. Pick arbitrary $X(x,y,z)$ in our plane. Then, the equation of our plane is given by $$ \vec{N} \cdot \vec{PX} = 0 \iff ...


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I talked to some people outside the internet and one interpretation is as follows: We have that if $T = {\bf x}\times$, then $T$ fixes the line spanned by ${\bf x}$, and since $T$ is anti-symmetric, $T$ leaves fixed the complement of that line: the plane normal to ${\bf x}$. The restriction of $T$ to that plane ${\bf x}^\perp$ works as a rotation of $90$ ...


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The essentials are getting a bit lost in the coordinate forms you're using. There's no nice way to write this in coordinates because there are no canonical elements of the eigenspaces. It gets a lot clearer if you abstract from the coordinates. Take any vector $\mathbf y$ orthogonal to $\mathbf x$. Then $$ \mathbf y+\mathrm i\frac {\mathbf x}{\|\mathbf ...


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Here is a fully geometric view, showing in particular 1) the deep analogy of operator $\left[n_{\times}\right]$ with its eigenvalues. 2) the fact that, using Cayley-Hamilton theorem, $$\left[n_{\times}\right]^3=-\left[n_{\times}\right]$$


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We have to pick one way to do it - either the right hand rule or the left hand rule. It's not important which one we use, as long as everyone uses the same one - just like which side of the road we drive on. We'd get the same eventual results if we all used the left hand rule. Someone chose the right hand rule over the left hand rule, and it became ...


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It is the way the cross product and the convention of the coordinate system is defined. https://en.wikipedia.org/wiki/Right-hand_rule#Coordinates



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