# How do we find out whether four points #A(3,-1,-1),B(-2,1,2)#, #C(8,-3,0)# and #D(0,2,-1)# lie in the same plane or not?

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To determine if the points ( A(3, -1, -1) ), ( B(-2, 1, 2) ), ( C(8, -3, 0) ), and ( D(0, 2, -1) ) lie in the same plane, we can check if the vectors formed by connecting any three of the points are coplanar. If they are coplanar, then all four points are in the same plane. Otherwise, they are not.

We can use the cross product to check coplanarity. If the cross product of two vectors is zero, then they are parallel, and the points lie on the same plane. We'll choose three vectors formed by the points and calculate their cross products:

- ( \vec{AB} = B - A )
- ( \vec{AC} = C - A )
- ( \vec{AD} = D - A )

Then, we'll calculate the cross products of pairs of these vectors:

- ( \vec{AB} \times \vec{AC} )
- ( \vec{AB} \times \vec{AD} )

If both cross products are zero, the points are coplanar; if not, they are not coplanar.

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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.

- What is the perimeter of a triangle with corners at #(7 ,3 )#, #(4 ,5 )#, and #(3 ,3 )#?
- Circle A has a center at #(-5 ,8 )# and a radius of #4 #. Circle B has a center at #(-3 ,3 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?
- An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(2 ,5 )# to #(8 ,7 )# and the triangle's area is #12 #, what are the possible coordinates of the triangle's third corner?
- What is the perimeter of a triangle with corners at #(3 ,7 )#, #(1 ,3 )#, and #(4 ,9 )#?
- Circle A has a center at #(-1 ,2 )# and a radius of #2 #. Circle B has a center at #(5 ,-4 )# and a radius of #1 #. Do the circles overlap? If not what is the smallest distance between them?

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