2 interactive labs · AS & A Level Mathematics 9709
Lines in 3D: Parallel, Intersecting or Skew
3.7 Vectors · Set two lines r = a + λb in space, predict parallel, intersect or skew, then equate components, solve two equations and check the third.
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About this lab
Set two lines r = a + λb in space, predict parallel, intersect or skew, then equate components, solve two equations and check the third.
You will be able to
State the meaning of a, b and the parameter in r = a + λb, and write the equation of a line in column-vector or i, j, k form.
Determine whether two lines are parallel by testing whether one direction vector is a multiple of the other, and whether parallel lines coincide.
Calculate λ and μ by equating components, solving two equations and checking the third, then state the position vector of the point of intersection.
Deduce that lines which are not parallel and have no common point are skew, and explain why a flat picture cannot show it.
The Scalar Product: Angles and Feet of Perpendiculars
3.7 Vectors · Pick two vectors, a cuboid, or a point and a line, predict the angle or the foot, then work it out from a·b = |a||b| cos θ and PF·d = 0.
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About this lab
Pick two vectors, a cuboid, or a point and a line, predict the angle or the foot, then work it out from a·b = |a||b| cos θ and PF·d = 0.
You will be able to
Calculate a·b from components, the magnitudes |a| and |b|, and the angle between the vectors from cos θ = a·b / |a||b| to 1 d.p., including angles in a cuboid.
Deduce from the sign of a·b whether the angle is acute, a right angle or obtuse, and that a·b = 0 means the vectors are perpendicular.
Determine the acute angle between two lines from their direction vectors, and explain why reversing a direction vector gives the same lines.
Calculate the foot of the perpendicular from a point to a line by solving PF·d = 0, and the perpendicular distance |PF|.
