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#### • collinear points

From Wikipedia

Orbital plane (astronomy)

All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded. Three non-collinear points in space suffice to define the orbital plane. A common example would be: the center of the heavier object, the center of the orbiting object and the center of the orbiting object at some later time.

By definition the inclination of a planet in the solar system is the angle between its orbital plane and the orbital plane of the Earth (the ecliptic). In other cases, for instance a moon orbiting another planet, it is convenient to define the inclination of the moon's orbit as the angle between its orbital plane and the planet's equator.

## Artificial satellites around the Earth

For launch vehicles and artificial satellites, the orbital plane is a defining parameter of an orbit; as in general, it will take a very large amount of propellant to change the orbital plane of an object. Other parameters, such as the orbital period, the eccentricity of the orbit and the phase of the orbit are more easily changed by propulsion systems.

Orbital planes of satellites are perturbed by the non-spherical nature of the Earth's gravity, and this causes the orbital plane to slowly rotate around the Earth, depending on the angle of the plane. For planes that are at a critical angle this can mean that the plane will track the Sun around the Earth, forming a Sun-synchronous orbit.

A launch vehicle's launch window is usually determined by the times when the target orbital plane intersects the launch site.

Table cell

A table cell is one grouping within a table. Cells are grouped horizontally (rows of cells) and vertically (columns of cells). Usually information on the top header of a table and side header will "meet" in the middle at a particular cell with information regarding the two headers it is collinear with.

## HTML Usage

### Kinds of cells in HTML

A table cell in HTML is a non-empty element and should always be closed. There are two different kinds of table cells in HTML: normal table cells and header cells. &lt;td&gt; denotes a table cell, while &lt;th&gt; denotes a table header. The two can be used interchangeably, but it is recommended that header cells be only used for the top and side headers of a table.

### Syntax

A table cell also must be nested within a &lt;table&gt; tag and a &lt;tr&gt; (table row) tag. If there are more table cell tags in any given row than in any other, the particular &lt;tr&gt; must be given a colspancell declaring how many columns of cells wide it should be.

### Example

An example of an HTML table containing 4 cells:

 Cell 1 Cell 2 Cell 3 Cell 4
HTML source:
 Cell 1 Cell 2 Cell 3 Cell 4

### Colspan and Rowspan

Every row must have the same number of table data cells, occasionally table data cells have to span more than one column or row. In this case the tags colspan and/or rowspan are used - where they are set to a number.

Question:What is collinear? How can a set of POINTS all be collinear? I have this example; "Show that A (4,11) B (6,8) and C (14,14) are collinear." I worked out so far that AB has the same gradient as BC but how does that prove that all these points all sit on the same line and why is B a common point? Thank you!

Answers:collinear points all sit on a COmmon LINE. the line through AB has slope (gradient) -3/2 the line through BC has slope 6/8 = 3/4 the line through AC has slope 3/10 so they are NOT collinear. the equation of the line through A and B is y = (-3/2)x + 17, and so (14,-4) is on the line, but not (14,14) unless you mistyped something.

Question:Give an example of three distinct points collinear to P = (1; 2; 3) and Q = (3; 2; 1). First question, is this question asking for 3 points collinear to P and 3 collinear to Q? Or is it asking for 3 points collinear to line PQ. Cant I find collinear points by multiplying the vector by some constant c? Jeff, do you mean (2,0,-2)? That's the difference between Q and P. Or are you factoring 2 out of the vector?

Answers:Any point on that line is: (1,2,3) + k(1,0,-1) for any real number k For example, for P, k = 0, and for Q, k = 2 So here are some other points on that line: If k = 1, we have (2,2,2) If k = 4, we have (5,2,-1) If k = 5, we have (6,2,-2) etc.

Question:Is there a way to tell, without graphing the points themselves if 3 sets of coordinates would be collinear? How would you go about doing so? Would you use y=mx+b format? Thank you!

Answers:Use the slope formula. If the difference of the y's divided by the difference of the x's is the same for two pairs of points, then they are collinear. examples: (2,5) (0,1) (15,31) First pair: (5-1)/(2-0) = 4/2=2 Second pair: (1-31)/(0-15)=-30/-15=2 Both pairs have the same slope, so the lines are collinear (You can also use the third pair: (31-5)/(15-2)=26/13=2) Contra-example: (-1,1), (-2,2), (-3,0) First pair: (1-2)/(-1+2)=-1/1=-1 Second pair: (2-0)/(-2+3)=2/1=2 If you connect the dots of the first pair, the line has a slope of -1. If you connect the dots of the second pair, the line has a slope of 2. These points can't be collinear. _/

Question:I have to take pictures of things in everyday life that represent these geometry subjects. They are: Acute Angle Adjacent Angle Angle Bisector Collinear Concave Polygon Congruent Convex Polygon Line Line Segment Linear pair Midpoint Obtuse Angle Parallel lines Perpendicular Lines Plane Point Ray Right Angle Slope Vertical Angles If you could help me by giving examples of what in the real world could represent any of these, that would be great. I have a couple but I just want to make sure that I am good on them. Thank you. =]

Answers:Acute Angle : find something triangular. All triangles have at least one acute angle Adjacent Angle : take that same triangular thing and any 2 angles are adjacent. Angle Bisector : Collinear find something with a straight line and any 3 (or more) things along that line are collinear Concave Polygon Congruent: get two things that are identical in size Convex Polygon Line : use a yardstick and mention it goes on forever. Route 66 goes across the country, but it is not straight Line Segment : use a ruler Linear pair Midpoint: the number 6 on a ruler Obtuse Angle; get a triangle again Parallel lines; find a box and two of the edges will be parallel Perpendicular Lines ; use the same box and use two edges that are perpendicular Plane ; a piece of paper could represent a plane surface, understand it goes on in all directions Point; has no dimensions, but for school purposes, a dot Ray: the graph of the absolute value of x Right Angle: anything that meets at a right angle. Use the box again Slope: a hill Vertical Angles: a map where 2 roads cross good luck