Further vectors

What further vectors covers

Straight lines and flat planes written as equations, and the geometry between them computed by the scalar product: meeting points, angles and shortest distances. Core Pure content, examinable on both compulsory papers of 9FM0. This is where the scalar product is introduced, since the A level Mathematics vectors unit stops before it.

The main ideas

• A line as r = a + lambda b, one anchor point plus multiples of a direction, converted to cartesian form and back.
• A plane in scalar product form and in cartesian form, with a normal found by making it perpendicular to two directions in the plane.
• Deciding whether two lines meet, are parallel, or are skew.
• Where a line meets a plane, found by substituting the parametric coordinates and solving one equation in the parameter.
• Angles: a sine with the normal for a line and a plane, a cosine of the two normals for two planes.
• Perpendicular distance from a point to a plane and to a line, and the shortest distance between two skew lines.

The results it turns on

r = a + λb

a line through the point a in the direction b

r·n = d, and ax + by + cz = e

the two forms of a plane, with n the vector (a, b, c)

a·b = |a||b| cos θ

the scalar product, with a zero value meaning perpendicular

|ax₁ + by₁ + cz₁ − d|/√(a² + b² + c²)

the perpendicular distance from a point to a plane

sin of the line-plane angle = |b·n|/(|b||n|)

the angle between a line and a plane, using the normal

Where it usually goes wrong

• Two equations for the same line can look completely different, because the anchor point may be any point on it and the direction may be scaled. Both earn the same marks.
• A cartesian line equation with a zero denominator means that coordinate is constant, and it is written as a separate statement rather than as a fraction.
• If the direction is perpendicular to the normal the line is parallel to the plane, and testing a single point separates parallel from lying inside it.
• For a shortest distance the perpendicularity conditions are where the method marks sit, so write them down before solving.

Where to start

Lines and planes first, and spend the time on identifying direction vectors and normals, since most of the later marks hang on picking the right one. Intersections, angles and distances second. Revise the A level vectors unit beforehand, because the notation carries straight over.