Prove that if two lines are each perpendicular to a third line in the plane, then the two lines are parallel.

Q: Verify that the two lines are parallel and find the distance between the lines. L1:   x = 2 − t,   y…

A:

Q: Give an example of three planes,exactly two of which are parallel.

A: Give an example of three planes,exactly two of which are parallel.

Q: Show that the lines determined byr 1(t) = <−2 − 5t, 3 + 2t, 4t >andr 2(t) = <8 + 15t, 1 −…

A:

Q: Prove the following theorems: 1. In a plane, there is exactly one line perpendicular to a given line…

A: The objective of the given problem is: "Prove that in a plane, there is exactly one line…

Q: Show that the lines having the equations 4x – 3y + 12 = 0 and 8x – 6y + 15 = 0 are parallel.

A: According to question,Given equation of lines,4x - 3y +12 = 08x - 6y +15 =0

Q: Any two lines in space are either intersecting or parallel.

A: Given statement is  ' Any two lines in space are either intersecting or parallel' . The statement is…

Q: If there were no understood restriction to lines in a plane in Theorem 1.4.3, the theorem would be…

A: The given statement is " In space, there is exactly one line perpendicular to a given line at any…

Q: 11a) Are the two lines parallel? Show work 4 (3, 2) -30] 2. (5,-1) 【け1.-B12 2

A: Given: Co-ordinates of line l1=(-3,0) and (3,2) Co-ordinates of line l2=(-1,-3) and (5,-1)

Q: Determine if the statements below are True or False and Explain in detail. 1. If L is a line with…

A: (1) True Since L and P do not intersect, that means they are parallel This implies that d will be…

Q: Determine the symmetric equations of line that passes through the points B and С.

A:

Q: would be on the perpendicular transversal if its

A: Given,        y=43x+13

Q: Show that the lines L1 : x =1– t, y = 2t, z = 4+t and L2 : x = 1+ 3t, y= 3 – 6t, z = 5 – 3t are…

A:

Q: Postulate 12 states that if two points lie in a plane, the line that contains them lies in the…

A: We need to determine whether the given statement is true or false. Postulate 12 states that if two…

Q: Show that the lines 2 x+3 y-1 and 6 x-12 y-1=0 are not perpendicular

A:

Q: Prove that the point (1,11),(2,15), and (-3,-5) are collinear and find the equation of the straight…

A: Let the given points be A (1,11), B (2,15), and C (-3,-5). Evaluate the slope of the line AB as…

Q: 1. How can two pencils (representing lines) be held so that a plane can be passed containing both;…

A: Please submit parts 2 and 3 separately for them to be answered. For the first part, Two lines, in…

Q: Every tow different lines only intersect at one point

A: If a line intersect at more than one point then it coinside

Q: The distanced of a point P to the line through points A and B is the length of the component of AP…

A: Answer

Q: The set of all points in a plane that are equidistant from a fixed line and a fixed point is a/an…

A: Parabola is defined as a curve made up of all points that are equidistant from a fixed line and a…

Q: Prove that the sum of the squares of the lengths of the sides of a parallelogram is equal to the sum…

A: Let the sides of the parallelogram be 'l' with longest and 'b' with shortest parallel lines and…

Q: Determine whether the line x=3t ,y=1+t ,z=2t  and the plane x-y-z= 1 are parallel. If they are…

A:

Q: In each case, determine whether or not the lines have a single point of intersection. If they do,…

A:

Q: Prove that if two nonvertical lines have slopes whose product is - 1, then the lines are…

A: As we are given that the product of slopes is -1let l1 be the first non-vertical line having slope…

Q: For every two lines intersecting in space, there is only one plane in them True false

A:

Q: Depict the relationship of Parallel Postulate and Euclidean  Geometry.

A: Let's write answer,

Q: Which of the following represents two lines that are coplanar and do not intersect?

A: We want to find the lines which are coplanar and don't intersect.

Q: Assume that all lines that appear tar 1. Find AB

A: We can find the length of AB in the following step

Q: Write the characterisation of two parallel lines in terms of the cross product and prove it

A:

Q: Prove analytically that if the diagonals of a parallelogram rectangle. are equal, then the…

A: Given:-  Diagonals of parallelogram are equal.To proof  Parallelogram is a Rectangle.

Q: if 2 lines are parallel,  then the rays that bisect a

A: It is given that the two lines are parallel . To prove : The rays that bisect a pair of…

Q: If VW is not parallel to UX, what is VW?

A: Clearly from the figure we can see that the base angles are same. So this figure is an isoceles…

Q: Draw and describe the locus of points in the plane that are equidistant from parallel lines l and m.

A: Recall the fact that locus maintain an equal distance from each lines. That is, locus must be half…

Q: Prove without using the concept of duality: In the four-line geometry, there exists a pair of points…

A:

Q: 1 Identify the statement as true or false and justify your answer. If m is perpendicular to plane T…

A:

Q: Prove that a line cannot contain three distinct points.

A: We prove that a line cannot contain three distinct points by method of contradiction.

Q: Consider any eight points such that no three are collinear.How many lines are determined?

A: Given :                         There are 8 points    To find :                      To…

Q: Prove that there exists a set of two lines containing all the points of the geometry.

A:

Q: Construct the line through C parallel to AB. A

A: The given triangle is △ABC. To draw a line passing through C and parallel to AB, Scribe an…

Q: D F OF and GI are parallel lines and mZDEH = 116°, %3D Submit

A: DF and GI are parallel line,

Q: Find the coordinates of the intersection point of the lines x_1 + 3x_2 = 3 and −2x_1 + 4x_2 = 1 in…

A: Solved below

Q: 3. What are the three possibilities for the intersection of a line and a plane?

A: The solution is given in the next step.

Q: Prove that, if two lines are parallel to a third line, then the lines are parallel to each other

A: Consider the following figure.

Q: Prove that parallel lines determine only one plane.

A:

Q: Prove that the line joining the midpoint of a median to a vertex of the triangle trisects the side…

A: Here ask to prove the line joining the midpoint of a median to a vertex of the triangle trisectsthe…

Q: Do two points determine a plane?

A: No.

Q: Suppose the symmetric equations of lines l, and l2 are = 2 – y = z and x = -1, = respectively.

A:

Q: . Name five ways to prove that two lines are parallel.

A: Name five ways to prove that two lines are parallel.

Q: Do two intersecting lines determine a plane?

A: Two lines intersect at only one point.

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.