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What is the formula for the cosine of the angle θ between two lines defined by r = ar1 + λ b1 and r = a2 + λ b2?
What is the formula for the cosine of the angle θ between two lines defined by r = ar1 + λ b1 and r = a2 + λ b2?
cos θ = (b1 ⋅ b2) / (|b1| |b2|)
If two lines are represented by their direction ratios l1, m1, n1 and l2, m2, n2, what is the formula for the acute angle between them?
If two lines are represented by their direction ratios l1, m1, n1 and l2, m2, n2, what is the formula for the acute angle between them?
cos θ = |l1 l2 + m1 m2 + n1 n2|
What is the shortest distance between two skew lines defined by the equations r = ar1 + λ b1 and r = a2 + µ b2?
What is the shortest distance between two skew lines defined by the equations r = ar1 + λ b1 and r = a2 + µ b2?
Shortest distance = |(b1 × b2) ⋅ (a2 – a1)| / |b1 × b2|
How can the shortest distance between two lines in three-dimensional space be represented?
How can the shortest distance between two lines in three-dimensional space be represented?
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What is the formula to calculate the distance between two parallel lines in three dimensions?
What is the formula to calculate the distance between two parallel lines in three dimensions?
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What are direction cosines?
What are direction cosines?
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If a directed line makes angles α, β, and γ with the x, y, and z-axes, respectively, what are these angles called?
If a directed line makes angles α, β, and γ with the x, y, and z-axes, respectively, what are these angles called?
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What is required for unique direction cosines of a directed line?
What is required for unique direction cosines of a directed line?
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Any three numbers which are proportional to the direction cosines of a line are called the direction __________.
Any three numbers which are proportional to the direction cosines of a line are called the direction __________.
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What is the relationship between direction cosines and direction ratios?
What is the relationship between direction cosines and direction ratios?
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What are the equations that give the direction cosines of a line passing through points P(x1, y1, z1) and Q(x2, y2, z2)?
What are the equations that give the direction cosines of a line passing through points P(x1, y1, z1) and Q(x2, y2, z2)?
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What is the direction cosine of a line making angles 90°, 60°, and 30° with the axes?
What is the direction cosine of a line making angles 90°, 60°, and 30° with the axes?
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How can one determine if three points A, B, and C are collinear?
How can one determine if three points A, B, and C are collinear?
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What are the parameterized equations of a line through a given point A and parallel to a given vector?
What are the parameterized equations of a line through a given point A and parallel to a given vector?
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What is the angle formula between two lines L1 and L2 passing through the origin?
What is the angle formula between two lines L1 and L2 passing through the origin?
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What indicates that two lines are parallel?
What indicates that two lines are parallel?
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What is the Cartesian equation of a line?
What is the Cartesian equation of a line?
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What is the formula for the cosine of the angle θ between two lines with direction ratios a1, b1, c1 and a2, b2, c2?
What is the formula for the cosine of the angle θ between two lines with direction ratios a1, b1, c1 and a2, b2, c2?
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What is the shortest distance between two intersecting lines?
What is the shortest distance between two intersecting lines?
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What are skew lines?
What are skew lines?
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What is the condition for two lines to be parallel in three-dimensional geometry?
What is the condition for two lines to be parallel in three-dimensional geometry?
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What is the formula for calculating the shortest distance between two skew lines?
What is the formula for calculating the shortest distance between two skew lines?
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How can you determine the angle between two lines given by their direction cosines?
How can you determine the angle between two lines given by their direction cosines?
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What do direction cosines represent?
What do direction cosines represent?
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What is the relationship between direction cosines and direction ratios?
What is the relationship between direction cosines and direction ratios?
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In vector form, how can you represent a line passing through a point with position vector ar and parallel to vector b?
In vector form, how can you represent a line passing through a point with position vector ar and parallel to vector b?
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What is the Cartesian equation of a line given by direction ratios a, b, and c through a point?
What is the Cartesian equation of a line given by direction ratios a, b, and c through a point?
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Study Notes
Introduction to Three Dimensional Geometry
- Vector algebra is used to simplify the study of 3-dimensional geometry.
- This chapter covers topics like: direction cosines and ratios of a line, equations of lines and planes, angles between lines and planes, shortest distance between skew lines, and distance of a point from a plane.
Direction Cosines and Direction Ratios
- Direction cosines are cosines of angles a line makes with the x, y, and z-axes (α, β, γ, respectively).
- Direction ratios are any three numbers proportional to the direction cosines of a line.
- If l, m, n are direction cosines and a, b, c are direction ratios, then: a = λl, b = λm, c = λn, where λ ≠ 0.
- The direction cosines can be found using the formula: l = ± a/(√(a² + b² + c²)), m = ± b/(√(a² + b² + c²)), n = ± c/(√(a² + b² + c²)) .
- There are infinitely many sets of direction ratios for a line.
Direction Cosines of a Line Passing Through Two points
- The direction cosines of the line segment joining points P(x1, y1, z1) and Q(x2, y2, z2) are:
- (x2 - x1)/PQ, (y2 - y1)/PQ, (z2 - z1)/PQ
- Where PQ = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)
Equation of a Line in Space
- A line is uniquely determined by:
- Passing through a given point and parallel to a given vector.
- Passing through two given points.
Equation of a Line Through a Given Point and Parallel to a Given Vector
- The vector equation of the line is: r = a + λb, where:
- r is the position vector of any point P on the line.
- a is the position vector of the given point A.
- b is the vector parallel to the line.
- λ is a real number.
- The Cartesian equation of the line is: (x - x1)/a = (y - y1)/b = (z - z1)/c, where:
- (x1, y1, z1) are the coordinates of point A.
- a, b, c are the direction ratios of the line.
Angle Between Two Lines
- The angle θ between two lines with direction ratios a1, b1, c1 and a2, b2, c2 is given by:
- cos θ = (a1a2 + b1b2 + c1c2)/√(a1² + b1² + c1²)√(a2² + b2² + c2²)
- Two lines are perpendicular if a1a2 + b1b2 + c1c2 = 0.
- Two lines are parallel if a1/a2 = b1/b2 = c1/c2.
Shortest Distance Between Two Lines
- If two lines intersect, the shortest distance between them is 0.
- If two lines are parallel, the shortest distance is the perpendicular distance between them.
- Skew lines are lines that are neither intersecting nor parallel. The shortest distance is not directly on either line but connects points on each line.
- The concept of shortest distance is crucial to understand the spatial relationships between lines.
Shortest distance between skew lines
- The shortest distance between two skew lines is perpendicular to both lines.
- The shortest distance vector will be equal to the projection of the line segment joining any two points on the lines (one point on each line) along the direction of the shortest distance vector.
- The unit vector n̂ along the shortest distance vector is given by n̂ = (b1 × b2) / |b1 × b2|, where b1 and b2 are the direction vectors of the lines.
- The shortest distance d is then calculated as d = |PQ| = |ST| |cos θ|, where ST is the line segment joining points on the lines and θ is the angle between ST and PQ.
- cos θ can be expressed as (b1 × b2) ⋅ (a2 − a1) / (|b1 × b2| |ST|), where a1 and a2 are the position vectors of the points on the lines.
- The final formula for the shortest distance is d = | (b1 × b2) ⋅ (a2 − a1) | / |b1 × b2|.
Shortest distance between parallel lines
- The shortest distance between two parallel lines is the distance between a point on one line and its foot perpendicular onto the other line.
- This distance is calculated as d = |b × (a2 - a1) / |b|, where b is the direction vector of the lines, and a1 and a2 are position vectors of points on the lines.
Cartesian form of shortest distance between skew lines
- The shortest distance between the lines (x − x1)/a1 = (y − y1)/b1 = (z − z1)/c1 and (x − x2)/a2 = (y − y2)/b2 = (z − z2)/c2 can be calculated using the formula: d = √((b1c2 − b2c1)² + (c1a2 − c2a1)² + (a1b2 − a2b1)²) / √((a1² + b1² + c1²) * (a2² + b2² + c2²)).
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Explore the fundamentals of three-dimensional geometry with this quiz. Dive into topics such as direction cosines, direction ratios, and the equations of lines and planes. Test your understanding of 3D concepts and improve your skills in vector algebra.
