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A line can either lie on a plane, lie parallel to it or intersect it. Determine, if there is one, the point of intersection between the line given by the equation

$$\displaystyle\frac{x−5}{2} =\displaystyle\frac{y−1}{-1} = \displaystyle\frac{z−15}{4}$$
and the plane given by the equation $$(x, y, z) = (-2, -7, 5) + s(2, 6, 3) + t(1, 4, -1)$$

So, what I have to do, is determine if the line and the plane either intersect or are parallel? What equation applies in this problem?

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  • $\begingroup$ Write the plane equation without parameters, i.e. $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$. Then use the line equation to solve for two of your variables in terms of the other and plug that in to the equation for your plane. $\endgroup$
    – Brent
    Mar 19, 2015 at 14:23
  • $\begingroup$ Can you calculate a vector that is parallel with line? Can you calculate vector perpendicular to the plane? $\endgroup$
    – zoli
    Mar 19, 2015 at 14:25
  • $\begingroup$ Hmmm, When you say write the plane equation without parameters, you mean: . a(2−5)+b(-1−1)+c(15−4)=0 ? $\endgroup$
    – Julio
    Mar 19, 2015 at 14:41

2 Answers 2

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A very basic way could be to find $x$, $y$ and $z$ from plane's equation and then put them into line's equation. This would lead us to see if the achieved system of equations is consistent or not: $$x=-2+2s+t, y=-7+6s+4t, z=5+3s-t$$

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By inspection we can find that the line is parallel to the vector $(2,-1,4)$ and that the point $(5,1,15)$ lies on the line. In homogeneous coordinates, the line is thus the join of $\mathbf p=[2:-1:4:0]$ and $\mathbf q=[5:1:15:1]$. This line can be represented by its Plücker matrix $\mathbf p\mathbf q^T-\mathbf q\mathbf p^T$.

A normal to the plane is $(2,6,3)\times(1,4,-1)=(-18,5,2)$ and $(-18,5,2)\cdot(-2,-7,5)=11$, so the plane can be represented in homogeneous coordinates as $\mathbf\pi=[-18:5:2:-11]$. The intersection of the line and plane is $$L\mathbf\pi=\mathbf p\mathbf q^T\mathbf\pi-\mathbf q\mathbf p^T\mathbf\pi=-66\mathbf p+33\mathbf q=[33:99:231:33]$$ or in (inhomogeneous) Cartesian coordinates, $(1,3,7)$.

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