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Allushta [10]
3 years ago
11

Find the equation for the plane that contains the line x=−1+3t , y=1+2t, z=2+4t and is perpendicular to the plane containing the

two lines
L1: x=2t+1,y=-t-1,z=6t+5

L2:x=t+1,y=t-1,z=-3t+5
Mathematics
1 answer:
Ivan3 years ago
8 0

Let L be the line given by the vector equation

(-1,1,2) + \lambda(3,2,4) \ , \lambda \in \mathbb{R}.

First, we use the director vectors of the lines L1 and L2 to get the

vector equation of the plane containing them, which we denote by \Pi_1. This is,

\\\\\Pi_1  : (1,-1,5) + \alpha (2,-1,6) + \beta (1,1,-3) \ , \alpha, \beta \in \mathbb{R}\\\\\\

We observe that \vec{N} = (2,-1,6)\times(1,1,-3) = (-3,12,3) \ne (0,0,0). Therefore, the vector equation of \Pi_1 defines a plane and \vec{N} is a normal vector to \Pi_1.

 

Finally, the vector equation for the wanted plane, which we denote by \Pi, is

\Pi : (-1,1,2) + r(3,2,4) + s(-3,12,3), r,s \in \mathbb{R} \ .

Thus, if s = 0, then L \subset \Pi and since \vec{N} is parallel to \Pi, then it is perpendicular to \Pi_1.

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