RI/2021/JC2/Prelim/P2/Q04

(a)(i)

Referred to the origin $$O$$, points $$A, B$$ and $$C$$ have position vectors a, $$b$$ and c respectively. The three points lie on a circle with centre $$O$$ and diameter $$A B$$ (see diagram).

Using a suitable scalar product, show that the angle $$A C B$$ is $$90^{\circ}$$.



(ii) The variable vector $$r$$ satisfies the equation $$( r – i ) \cdot( r – k )=0$$. Describe the set of vectors $$r$$ geometrically.



(b) (i) The variable vector $$r$$ satisfies the equation $$r \cdot n = m \cdot n$$, where $$m$$ and $$n$$ are constant vectors. Describe the set of vectors $$r$$ geometrically. Give the geometrical meaning of $$| m \cdot n |$$ if $$n$$ is a unit vector.



(ii) The plane $$\pi$$ passes through the points with position vectors $$x i, y j$$ and zk where $$x, y$$ and $$z$$ are non-zero constants. It is given that $$d$$ is the perpendicular distance from the origin to $$\pi$$. Show, by finding the normal of $$\pi$$, or otherwise, that $$\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{1}{d^{2}}$$.