Question

Prove True Fact 2: True Fact 1: If A-B-C and line L passes through B but not A, then A and C lie on opposite sides of L. TF1 is used to prove the following (in fact, the proof is not much different): True Fact 2: If point A lies on L and point B lies on one of the half-planes determined by L, then, except for A, the segment AB or ray AB lies completely in that half-plane.

Answer #1

**If point A lies on L and point B lies on one of the
half-planes determined by L, then, except for A, the segment AB or
ray AB lies completely in that half-plane.**

**PROOF:**

**Let point A line on the line L and point B be in H1.
Using the Ruler Postulate, we may find C such that A – C – B. Now C
is in H1, on the line L, or in H2..**

**Suppose C is on L. This puts B on L since C and B are
collinear by betweenness.**

**Thus we know that C is not on L.**

**Suppose C is n H1. Then, by Axiom D – 1, c. we have a
point E on L with**

**A – C – E – B, which would put B on L by betweenness.
Again, a contradiction.**

**So C is in H1.**

**A similar argument works for using the Ruler Postulate
to find point F such**

**that A – B – F.**

**This puts both the segment AB and the ray AB in
H1.**

**Q.E.D.**

prove that if C is an element of ray AB and C is not equal to A,
then ray AB = ray AC using any of the following corollarys
3.2.18.) Let, A, B, and C be three points such that B lies on
ray AC. Then A * B * C if and only if AB < AC.
3.2.19.) If A, B, and C are three distinct collinear points,
then exactly one of them lies between the other two.
3.2.20.)...

1. A plane passes through A(1, 2, 3), B(1, -1, 0) and
C(2, -3, -4). Determine vector and parametric equations of
the plane. You must show and explain all steps for full marks. Use
AB and AC as your direction vectors and point A as your starting
(x,y,z) value.
2. Determine if the point (4,-2,0) lies in the plane with vector
equation (x, y, z) = (2, 0, -1) + s(4, -2, 1) + t(-3, -1,
2).

Find an equation of the plane. The plane that passes through the
point (?1, 1, 1) and contains the line of intersection of the
planes x + y ? z = 2 and 4x ? y + 5z = 3

The line l1 has the direction vector h1,0,−1i and passes through
the point (0,−1,−1). The line l2 passes through the points (1,2,3)
and (1,3,2).
a. [2] What is the angle between l1 and l2 in radians? The
answer should lie between 0 and π/2.
b. [6] What is the distance between l1 and l2?

Three parallel lines are such that one passes through each
vertex of a triangle ABC, and they are not parallel to any of the
triangles sides. The line through A meets BC (extended if
necessary) in X, the lines through B and C meet CA and AB in Y and
Z respectively. Prove that area(XYZ) = 2xArea(ABC)

a) Let L be the line through (2,-1,1) and (3,2,2). Parameterize
L. Find the point Q where L intersects the xy-plane.
b) Find the angle that the line through (0,-1,1) and (√3,1,4)
makes with a normal vector to the xy-plane.
c) Find the distance from the point (3,1,-2) to the plane
x-2y+z=4.
d) Find a Cartesian equation for the plane containing (1,1,2),
(2,1,1) and (1,2,1)

1)Find sets of parametric equations and symmetric equations of
the line that passes through the two points. (For the line, write
the direction numbers as integers.)
(5, 0, 2), (7, 10, 6)
Find sets of parametric equations.
2) Find a set of parametric equations of the line with the given
characteristics. (Enter your answer as a comma-separated list of
equations in terms of x, y, z, and
t.)
The line passes through the point (2, 1, 4) and is parallel to...

(a) Find the volume of the parallelepiped determined by the
vectors a =< 2, −1, 3 >, b =< −3, 0, 1 >, c =< 2, 4,
1 >.
(b) Find an equation of the plane that passes through the point
(2, 4, −3) and is perpendicular to the planes 3x + 2y − z = 1 and x
− 2y + 3z = 4.

Define a+b=a+b -1 and a*b=ab-(a+b)+2 Assume that (Z, +,*) is a
ring. (a) Prove that the additative identity is 1? (b) what is the
multipicative identity? (Make sure you proe that your claim is
true). (c) Prove that the ring is commutative. (d) Prove that the
ring is an integral domain. (Abstrat Algebra)

12. Find the equation of the following planes:
(b) pass through the point (1,-1,-1) and parallel to the plane
5x − y − z = 6
. c) pass through the points (2,1,2), (3, -8, 6), and (-2, -3,
1)

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