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Chapter Analysis
Advanced10 pages • EnglishQuick Summary
This chapter on Three Dimensional Geometry explores concepts related to direction cosines and ratios, equations of lines and planes in space, angles between lines and planes, the shortest distance between skew lines, and distance from a point to a plane. The chapter emphasizes using vector algebra to simplify and analyze these three-dimensional geometrical concepts, translating results into both vector and Cartesian forms. This approach offers a more intuitive and geometric understanding of spatial relationships.
Key Topics
- •Direction cosines and direction ratios
- •Equations of lines and planes in space
- •Angles between lines and planes
- •Shortest distance between skew lines
- •Distance from a point to a plane
- •Vector algebra applications in geometry
- •Conversion between vector and Cartesian forms
Learning Objectives
- ✓Understand and calculate direction cosines and ratios
- ✓Derive equations of lines and planes in three-dimensional space
- ✓Determine angles between lines, lines and planes, and planes
- ✓Calculate shortest distances between skew lines
- ✓Apply vector algebra to solve three-dimensional geometry problems
- ✓Translate between vector forms and Cartesian coordinate forms
Questions in Chapter
Show that the three lines with direction cosines (12/13, 3/13, 4/13), (4/13, 12/13, 3/13), (3/13, 4/13, 12/13) are mutually perpendicular.
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Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
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Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector i j k.
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Find the cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by x/3 = y/4 = z/8.
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The cartesian equation of a line is x - 5/3 = (y - 4)/7 = (z - 6)/2. Write its vector form.
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Additional Practice Questions
Derive the Cartesian equation for a plane given three non-collinear points.
mediumAnswer: First, determine two vectors from the given points. Use the cross product to find a normal vector to the plane. The plane equation can be expressed as ax + by + cz = d, where (a, b, c) are the coordinates of the normal vector and d can be found by substituting the coordinates of any of the three points into the equation.
How is the direction cosine related to the direction ratios of a line in space?
easyAnswer: The direction cosines are the cosines of the angles made by the line with the coordinate axes. They are proportional to the direction ratios. If l, m, n are direction cosines and a, b, c are direction ratios, then l : m : n = a : b : c.
What is the geometric interpretation of skew lines, and how is their shortest distance calculated?
mediumAnswer: Skew lines are lines that do not intersect and are not parallel, thus existing in different planes. The shortest distance between two skew lines is along a line perpendicular to both. This is calculated using the vector cross product method.
Explain the significance of the angle between two planes.
hardAnswer: The angle between two intersecting planes is the angle between their normal vectors. This can be calculated using the dot product of the normal vectors divided by the product of their magnitudes.
How do you determine if two lines in space are parallel using their vector equations?
mediumAnswer: Two lines are parallel if their direction vectors are scalar multiples of each other. By representing the lines in vector form r1 = a + λb and r2 = c + µd, the lines are parallel if b = kd for some scalar k.