Paper 1, Section II, C

Algebra and Geometry | Part IA, 2007

(i) Describe geometrically the following surfaces in three-dimensional space:

(a) ru=αr\mathbf{r} \cdot \mathbf{u}=\alpha|\mathbf{r}|, where 0<α<10<|\alpha|<1

(b) r(ru)u=β|\mathbf{r}-(\mathbf{r} \cdot \mathbf{u}) \mathbf{u}|=\beta, where β>0\beta>0.

Here α\alpha and β\beta are fixed scalars and u\mathbf{u} is a fixed unit vector. You should identify the meaning of α,β\alpha, \beta and u\mathbf{u} for these surfaces.

(ii) The plane nr=p\mathbf{n} \cdot \mathbf{r}=p, where n\mathbf{n} is a fixed unit vector, and the sphere with centre c\mathbf{c} and radius aa intersect in a circle with centre b\mathbf{b} and radius ρ\rho.

(a) Show that bc=λn\mathbf{b}-\mathbf{c}=\lambda \mathbf{n}, where you should give λ\lambda in terms of aa and ρ\rho.

(b) Find ρ\rho in terms of c,n,a\mathbf{c}, \mathbf{n}, a and pp.

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