What are the equations of the planes that are parallel to the plane #x+2y-2z=1# and two units away from it?

Answer 1

#x+2y-2z+5=0# and #x+2y-7=0#

First we'll find the equation of ALL planes parallel to the original one. As a model consider this lesson:

Equation of a plane parallel to other

The normal vector is: #vec n=<1,2-2>#
The equation of the plane parallel to the original one passing through #P(x_0,y_0,z_0)# is:
#vec n*"< "x-x_0,y-y_0,z-z_0> =0# #<1,2,-2>*"<"x-x_0,y-y_0,z-z_0> =0# #x-x_0+2y-2y_0-2z+2z_0=0# #x+2y-2z-x_0-2y_0+2z_0=0#

Or

#x+2y-2z+d=0# [1] where #a=1#, #b=2#, #c=-2# and #d=-x_0-2y_0+2z_0#

Now we'll find planes that obey the previous formula and at a distance of 2 units from a point in the original plane. (We should expect 2 results, one for each half-space delimited by the original plane.) As a model consider this lesson:

Distance between 2 parallel planes

In the original plane let's choose a point. For instance, when #x=0# and #y=0#: #x+2y-2z=1# => #0+2*0-2z=1# => #z=-1/2# #-> P_1 (0,0,-1/2)#
In the formula of the distance between a point and a plane (not any plane but a plane parallel to the original one, equation [1] ), keeping #D=2#, and #d# as #d# itself, we get:
#D=|ax_1+by_1+cz_1+d|/sqrt(a^2 + b^2 + c^2)# #2=|1*0+2*0+(-2)*(-1/2)+d|/sqrt(1+4+4)# #|d+1|=2*3# => #|d+1|=6#
First solution: #d+1=6# => #d=5# #-> x+2y-2z+5=0#
Second solution: #d+1=-6# => #d=-7# #-> x+2y-2z-7=0#
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Answer 2

The equation of a plane parallel to the plane ( x + 2y - 2z = 1 ) and two units away from it is given by:

[ x + 2y - 2z = d ]

where ( d ) is the distance from the original plane. Since the new plane is two units away, ( d = 1 + 2 = 3 ). Thus, the equation of the plane is:

[ x + 2y - 2z = 3 ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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