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In 3D two lines can be parallel, intersect at a point, or be skew. This vectors topic covers testing which case applies and finding any point of intersection. It appears across A-Level and iAL Pure Maths with the major exam boards. Work through it free on eClassroom, with clear worked examples and instantly-marked questions that build your confidence for the exam.
How you do it
If the direction vectors are multiples of each other the lines are parallel. Otherwise equate the two equations and solve two of the three components for the parameters, then check them in the third. If it fails, the lines are skew.
Before you start
Best tackled after vector equations of lines. This sits in Pure 4. It leads on to 3D geometric problems.
What you learn
In three dimensions a pair of straight lines can be related in one of three ways. They may be parallel, they may intersect at a single point, or they may be skew — neither parallel nor intersecting, a situation that can only happen in 3D. (If the lines are parallel and share a point, they are the same line.) To test quickly, compare the components in the same order. Two lines are skew if they are not parallel and they do not intersect. If two lines are known to intersect, their three equations are consistent. The course works through parallel lines, do two lines intersect, and where?, skew lines, classifying a pair of lines, finding an unknown value and putting it together. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
In our A-level maths courses, Parallel, Intersecting & Skew Lines belongs to Vectors and is met in the second year of A-level. It runs to 6 concepts, each followed by a check, with 6 worked examples along the way. Related topics students usually take next include Scalar (Dot) Product, Vector Equations of Lines and Angles & Geometric Problems . Endless extra practice on this strand is available on the Vectors revision engine, which generates a fresh question each time.
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