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I have a test today in this geometry course, and this was on our review sheet if you could please help me out with this question it would be very helpful thank you

Distance in $\mathbb{R^3}$ from points $P=(x_1, y_1, z_1)$ and $Q=(x_2, y_2, z_2)$ is given by $|PQ| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

b) Describe the set of points equidistant to two points in $\mathbb{R^3}$. (So i tried to give this a try, and i said it would be a line but it doesnt make sense for some reason

c) Show that an isometry of $\mathbb{R^3}$ is determined by where it sends 4 points $A, B, C, D$ as long as $A, B, C, D$ are not coplanar (Hint:Do not lay on a single plane in $\mathbb{R^3}$)

A reflection in $\mathbb{R^3}$ is in a plae, for example, the reflection in in the $xy$-plane is given by

$r_{xy}(x, y, z) = (x, y, -z)$.

e) Show that any isometry of $\mathbb{R^3}$ is the composition of up to four reflections. A reflection in $\mathbb{R^3}$ is in a plane.

f)A screw transformation is a rotation about a line followed by a translation along that line ( think of turning a screw!) describe the reflections needed to produce a screw transformation alone a line $L \subset \mathbb{R^3}$. (I have no clue what this is talking about)

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For (a), you can do it by manipulating the distance formula. But call the two points $A$ and $B$. Note that the points equidistant from $A$ and $B$ are all the points in the plane that is perpendicular to $AB$ and passes through the midpoint of the line segment $AB$. –  André Nicolas Feb 5 '13 at 16:43
    
b) In $\Bbb R^2$ it would be the bisector line in between the 2 points, in $\Bbb R^3$ it is an orthogonal plane at the midpoint of the given segment. --- c) It is not a "Hint" it is the meaning of the word 'coplanar'. ---- f) They thought about showing how e) works in practice. –  Berci Feb 5 '13 at 16:44
    
i am sorry but how does that help me :s –  MathGeek Feb 5 '13 at 17:14
    
I'm guessing here. Are you confused about the meaning of the term equidistant perhaps? –  Glen The Udderboat Feb 5 '13 at 18:20
    
no i just dont know how to do this question, apparently it would be on my test today so i really need like a good answer –  MathGeek Feb 5 '13 at 18:43
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