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Chapter 2: Problem 2

Explain why you need at least three non collinear points to determine a plane.

Short Answer

Expert verified
At least three non-collinear points are required to determine a plane because any two points can form a line, but we need a third point that doesn't lie on this line in a three-dimensional space to completely and uniquely define a plane.

Step by step solution

01

Understanding Collinearity

Collinear points are the ones that lie in the same straight line. If points are collinear, you can draw a single straight line through all of them.
02

Understanding 3D Space

A plane is a flat, two-dimensional surface that extends infinitely far. It can be thought of as a floor or wall in regular space, but it's not bounded - it extends forever in all directions.
03

Visualizing the plane using points

Two distinct points in space can always form a straight line. In 3D space, if you try to add another point to define a plane, and if this point falls on the same line, that's still considered as being on the same line, not enough to define a full plane. You have to go 'outside' of this line, that is, to a third point which doesn鈥檛 lie on this line (so it's non-collinear), to fully determine a flat plane in the space.
04

Concluding

Therefore, in order to fully define a plane in a 3D space, a minimum of three non-collinear points are required. These points will offer enough information (position and orientation) to unambiguously determine the plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Collinear Points
Collinear points are points that lie along the same straight line. Imagine placing three pins in a straight row on a piece of paper. These pins are collinear, as you can stretch a straight line across all of them.
  • Easy to visualize with at least two points. If a third point lines up, it simply remains on the same line.
  • Collinearity implies no variance in any additional dimension, meaning nothing new is introduced beyond a straight path.
In geometric terms, collinear points are significant due to their inability to create a plane. If we only have collinear points, they won鈥檛 provide sufficient information for two-dimensional space. They exist in a one-dimensional line, limiting the space in which you can work.
3D Space
In our world, we perceive a 3D space made of length, width, and height. When thinking about geometry in this context, a plane is a crucial concept. A plane is like an infinite sheet of paper extending forever.
  • A plane is two-dimensional, despite existing in 3D space.
  • Think of a plane as a perfectly flat surface with no thickness.
This infinite nature of a plane in 3D space is pivotal for geometric construction and analysis. Understanding this spatial dimension helps explain why merely having a straight line (such as with collinear points) is insufficient to define a plane. To visualize a plane, one must begin from a broader perspective that includes this third spatial dimension.
Non-Collinear Points
Non-collinear points are essential for defining a plane. With these, no single straight line can pass through all the points at once. This separation is key. When you have three non-collinear points:
  • You find a distinct orientation or tilting in space.
  • These points offer a boundary and direction that determine a plane鈥檚 unique position.
For example, if you place three pins in a triangular pattern on that same piece of paper, these are non-collinear. They define a specific, unambiguous section of space, which can be extended to form a plane. Such a configuration is essential for various geometry problems since it imparts depth and structural integrity to your spatial understanding, distinguishing a plane distinctly from a line.

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Most popular questions from this chapter

In Exercises \(25-32\) , name the property of equality that the statement illustrates. $$\text { if }\mathrm{m} \angle \mathrm{A}=29^{\circ} \text { and } \mathrm{m} \angle \mathrm{B}=29^{\circ}, \text { then } \mathrm{m} \angle \mathrm{A}=\mathrm{m} \angle \mathrm{B}$$

In Exercises \(33-36,\) rewrite the statements as a single bi conditional statement. (See Example \(5 . )\) If a polygon has four sides, then it is a quadrilateral. If a polygon is a quadrilateral, then it has four sides.

In Exercises \(17-24,\) write the conditional statement \(p \rightarrow q,\) the converse \(q \rightarrow p,\) the inverse \(\sim p \rightarrow \sim q\) , and the contrapositive \(\quad \mathrm{q} \rightarrow \sim \mathrm{p}\) in words. Then decide whether each statement is true or false. Let \(\mathrm{p}\) be "you are in math class" and let \(\mathrm{q}\) be "you are in Geometry."

In Exercises \(17-20,\) use the Law of Detachment to determine what you can conclude from the given information, if possible. If a point divides a line segment into two congruent line segments, then the point is a midpoint. Point \(\mathrm{P}\) divides \(\overline{\mathrm{LH}}\) into two congruent line segments.

MATHEMATICAL CONNECTION on the statement "If \(x^{2}-10=x | 2,\) then \(x=4 "\) be combined with its converse to form a true biconditional statement?
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