Vectors, lines and planes in three dimensions

Learning objectives

Calculate the vector product a × b, interpret its magnitude |a||b| sin θ as the area of the parallelogram spanned by a and b, and use the fact that it is perpendicular to both vectors with direction given by the right-hand rule

Write the equation of a plane in the forms r · n = d, ax + by + cz = d and r = a + λb + μc, and convert freely between them, obtaining a normal vector as the vector product of two direction vectors in the plane

Find the angle between two lines, between a line and a plane, and between two planes, using scalar products of direction vectors and normals and taking the acute angle

Calculate the perpendicular distance from a point to a line and from a point to a plane, and the shortest distance between two skew lines as |(a₂ − a₁) · (b₁ × b₂)| divided by |b₁ × b₂|

Determine whether two lines are parallel, intersecting or skew by comparing direction vectors and testing for a consistent pair of parameters, and find the line of intersection of two planes

Use the scalar triple product a · (b × c) to test whether three vectors are coplanar, the test being that the triple product is zero, and hence whether two non-parallel lines meet