Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 5A (Vectors and Matrices)

This is the plane through $a$ perpendicular to the nonzero normal vector $n$.

This is the affine plane through $b,d,f$, provided $d-b$ and $f-b$ are independent. The scalars $\lambda ,\mu$ are affine coordinates in those two directions.

This is the sphere with centre $c$ and radius $\rho >0$.

The cross product of the plane normals is proportional to $(1,-1,-1)$, soThe point on the intersection perpendicular to $m$ iswhich satisfies both plane equations and $u\cdot m=0$. Hence the line is $r\times m=u\times m$.

The equally inclined line has direction $d=(1,1,1)/\sqrt{3}$. The distance between the two skew lines is

The signed displacement of the sphere centre from the plane is $g\cdot n-p$. Orthogonal projection gives the circle centrePythagoras givesA real circle exists when $|p-g\cdot n|\le A$: equality gives tangency and radius zero, while strict inequality gives a genuine circle.