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Chapter Analysis
Intermediate10 pages • EnglishQuick Summary
This chapter introduces the concept of three-dimensional geometry, expanding on the two-dimensional systems previously learned. It covers the coordinate axes and planes, the method to identify the position of a point using three coordinates in space, and the division of space into octants. Further, it introduces the distance formula in a three-dimensional space, offering examples and problems for practice. The chapter effectively connects these concepts with practical examples and exercises to facilitate understanding.
Key Topics
- •Coordinate System in 3D
- •Coordinate Planes and Axes
- •Octants
- •Distance Formula in 3D
- •Collinearity of Points
- •Equation of Planes
Learning Objectives
- ✓Understand and identify the coordinate planes and axes in three dimensions.
- ✓Learn to calculate the distance between two points in a three-dimensional space.
- ✓Apply the concepts of octants to determine the position of points.
- ✓Solve practical problems using three-dimensional geometry concepts.
- ✓Relate three-dimensional geometry to real-world applications.
Questions in Chapter
A point is on the x-axis. What are its y-coordinate and z-coordinates?
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A point is in the XZ-plane. What can you say about its y-coordinate?
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Name the octants in which the following points lie: (1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (– 4, 2, –5), (– 4, 2, 5), (–3, –1, 6) (– 2, –4, –7).
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Fill in the blanks: (i) The x-axis and y-axis taken together determine a plane known as _______. (ii) The coordinates of points in the XY-plane are of the form _______. (iii) Coordinate planes divide the space into ______ octants.
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Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1) (ii) (–3, 7, 2) and (2, 4, –1) (iii) (–1, 3, –4) and (1, –3, 4) (iv) (2, –1, 3) and (–2, 1, 3).
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Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
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Verify the following: (i) (0, 7, –10), (1, 6, –6) and (4, 9, –6) are the vertices of an isosceles triangle. (ii) (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of a right-angled triangle. (iii) (–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.
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Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
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Additional Practice Questions
What is the role of the origin in the three-dimensional coordinate system?
easyAnswer: The origin is the point where the x, y, and z-axes intersect at (0, 0, 0). It serves as the reference point for determining the position of other points in space.
How can you determine the coordinates of a point if you know it lies in a specific octant?
mediumAnswer: By knowing the octant, you can determine the signs of the coordinates. Use its position relative to the coordinate planes to establish the value signs (positive or negative) in the coordinate triplet (x, y, z).
Explain how distance can be computed between two points in three-dimensional space.
mediumAnswer: The distance between two points (x1, y1, z1) and (x2, y2, z2) is given by the formula √((x2-x1)² + (y2-y1)² + (z2-z1)²)
Describe how a point in the XPYOT plane affects its coordinates?
mediumAnswer: If a point lies completely on a particular plane, one of its coordinates is zero. Specifically, in the XPYOT plane, the z-coordinate is 0.
What are the geometric interpretations of the x, y, and z coordinates in space?
hardAnswer: x, y, and z coordinates represent the distances from the planes parallel to the other two axes, forming a rectangular prism based on the three axes extended into space.