# How do you find parametric equations for the line through P-naught=(3,-1,1) perpendicular to the plane 3x+5y-7z=29?

{ (x = 3 + lambda 3), (y = -1+lambda 5), (z = 1 -lambda 7) :}#

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To find parametric equations for the line through ( P_0 = (3, -1, 1) ) perpendicular to the plane ( 3x + 5y - 7z = 29 ), we first find the normal vector to the plane. Then, we use this normal vector to determine the direction vector of the line. Finally, we use the given point ( P_0 ) to write the parametric equations of the line.

The normal vector to the plane ( 3x + 5y - 7z = 29 ) is ( \mathbf{n} = \langle 3, 5, -7 \rangle ).

The direction vector of the line perpendicular to the plane is the same as the normal vector to the plane, so ( \mathbf{d} = \langle 3, 5, -7 \rangle ).

Thus, the parametric equations of the line are:

[ x(t) = 3 + 3t ] [ y(t) = -1 + 5t ] [ z(t) = 1 - 7t ]

where ( t ) is a parameter.

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