Pure Mathematics · 3rd secondary · First Term

Algebra - Straight lines and planes in space — The equation of a straight line in space

About this lesson

This unit focuses on straight lines in three-dimensional space. It begins by defining the direction vector of a line, direction cosines, and direction ratios, and explains how to find a direction vector from two points. It then presents the vector, parametric, and Cartesian forms of the equation of a line in space, with worked examples and exercises. The unit also covers how to find the angle between two lines, conditions for parallel and perpendicular lines, and how to determine whether lines intersect or are skew. Finally, it includes methods for finding the distance from a point to a line. By the end of the unit, students should be able to write equations of lines in different forms, analyze relationships between lines (parallel, perpendicular, intersecting, skew), and compute angles and distances involving lines in space.

Main topics in this lesson

  • Direction vector, direction ratios, and direction cosines of a line in space
  • Vector, parametric, and Cartesian equations of a line in space
  • Angle between two lines
  • Parallel and perpendicular lines
  • Intersecting and skew lines
  • Distance from a point to a line

Key terms and vocabulary

  • direction vector
  • direction ratios
  • direction cosines
  • directed angles
  • vector form of the equation of a straight line
  • parametric equations
  • Cartesian equation
  • angle between two straight lines
  • parallel lines
  • perpendicular lines
  • skew lines
  • intersection point
  • distance from a point to a straight line

Questions and answers on Algebra - Straight lines and planes in space — The equation of a straight line in space - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 3rd secondary, First Term, with practice exercises for revision and exam preparation.

1

المعادلة الديكارتية للخط المستقيم الذي يمر بالنقطة (2، -1، 4) ومتجه اتجاهه (0، 1، -2) هي:

  • 1 (x-2)/0 = (y+1)/1 = (z-4)/-2
  • 2 (x-2)/1 = (y+1)/0 = (z-4)/-2
  • 3 (x-2)/1 = (y+1)/1 = (z-4)/1
  • 4 (x+2)/0 = (y-1)/1 = (z+4)/-2
2

متجها اتجاه خطين هما (2، 3، -1) و(1، -1، 1). الخطان:

  • 1 متعامدان
  • 2 متوازيان
  • 3 متقاطعان
  • 4 مائلان
3

الخط المستقيم الذي معادلته r = (1، 2، 3) + t(2، 0، -1) يقطع مستوى xy عند النقطة:

  • 1 (7، 2، 0)
  • 2 (1، 2، 0)
  • 3 (3، 2، 2)
  • 4 (-1، 2، 4)
4

إذا كان متجه اتجاه خط هو (0، 3، 4)، فإن طول متجه الاتجاه يساوي:

  • 1 5
  • 2 7
  • 3 25
  • 4 1
5

المعادلة الديكارتية للخط المستقيم المار بالنقطتين (0، 0، 0) و(1، 2، 3) هي:

  • 1 x/1 = y/2 = z/3
  • 2 x/3 = y/2 = z/1
  • 3 (x-1)/1 = (y-2)/2 = (z-3)/3
  • 4 x = y = z
6

متجها اتجاه خطين هما (1، 1، 0) و(0، 1، 1). قياس الزاوية بين الخطين يساوي:

  • 1 60°
  • 2 30°
  • 3 90°
  • 4 45°

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