1.determine whether the given line and the given planes are parallel:
1.1.x=1+t,y=-1-t,z=-2t and x+2y+3z-9=0.
1.2.<0,1,2>+t<3,2,-1> and 4x-y+2z+1=0.
2.
2.1.find parametric equations of the line that passes through the point p=(2,0,-1) and is parallel to the vector n=<2,1,3>.
2.2.find parametric equations of the line that passes through the points a=(1,2,-3) and b=(7,2,-4).
2.3.find parametric equations for the line intersection of the planes -5x+y-2z=3 and 2x-3y+5z=-7.
1.1. The plane is orthogonal to the vector .
The line has a direction vector .
It is parallel to the plane with a normal vector , if and only if .
But since , this line and this plane are not parallel.
1.2. The plane is orthogonal to the vector .
The line has a direction vector .
It is parallel to the plane with a normal vector , if and only if .
But since , this line and this plane are not parallel.
2.1.find parametric equations of the line that passes through the point p=(2,0,-1) and is parallel to the vector n=<2,1,3>.
2.2.find parametric equations of the line that passes through the points a=(1,2,-3) and b=(7,2,-4).
2.3.find parametric equations for the line intersection of the planes -5x+y-2z=3 and 2x-3y+5z=-7.
Let z=t be a parameter. Then
-5x+y=3+2z=3+2t, y=5x+3+2t
2x-3y=-5z-7, 2x-(5x+3+2t)=-5t-7
-3x=-3t-4
x=t+4/3, y=5(t+4/3)+3+2t = 7t+29/3, z=t. This is parametric equation for the line intersection of the given planes.
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