What are the equations of the planes that are parallel to the plane #x+2y-2z=1# and two units away from it?
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
Or
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
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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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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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- A line passes through #(5 ,1 )# and #(6 ,5 )#. A second line passes through #(1 ,3 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A triangle has corners at #(5 ,6 )#, #(2 ,7 )#, and #(3 ,4 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(3 ,2 )# and a radius of #6 #. Circle B has a center at #(-2 ,1 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?
- What is the equation of a line that is perpendicular to - x + 2y = 4 and passes through the point (-2,1)?
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