Practice section 4 of 6

(i) Find a vector equation of the line through the points $A$ and $B$ with position vectors $7\mathbf i+8\mathbf j+9\mathbf k$ and $-\mathbf i-8\mathbf j+\mathbf k$ respectively.

(ii) The perpendicular to this line from the point $C$ with position vector $\mathbf i+8\mathbf j+3\mathbf k$ meets the line at the point $N$. Find the position vector $N$ and the ratio of $AN:NB$.

(iii) Find a Cartesian equation of the line which is a reflection of the line $AC$ in the line $AB$.

(a) Interpret geometrically the vector equation $\mathbf r=\mathbf a+t\mathbf b$, where $\mathbf a$ and $\mathbf b$ are constant vectors and $t$ is a parameter.

(b) The points $P$, $Q$ and $R$ have position vectors $\mathbf p$, $\mathbf q$ and $\mathbf r$ respectively. The points $P$ and $Q$ are fixed and $R$ varies. Given that $\mathbf p$ is non-zero and $(\mathbf r-\mathbf p)\times\mathbf q=\mathbf0$, describe geometrically the set of all possible positions of the point $R$.

With reference to an origin $O$, the points $A$, $B$, $C$ and $D$ are such that $\overrightarrow{OA}=\mathbf a$, $\overrightarrow{OB}=\mathbf b$, $\overrightarrow{OC}=3\mathbf a-\mathbf b$ and $\overrightarrow{OD}=2\mathbf a+3\mathbf b$. The lines $BC$ and $AD$ intersect at point $X$ (see diagram).

(i) Express $\overrightarrow{OX}$ in terms of $\mathbf a$ and $\mathbf b$.

(ii) The point $Y$ on $BC$ is such that $BY:YC=2:11$. Show that the points $O$, $Y$ and $D$ are collinear.

(iii) Find the ratio of the area of triangle $OAB$ to the area of triangle $BCD$.

Electrical engineers are installing electricity cables on a building site. Points $(x,y,z)$ are defined relative to a main switching site at $(0,0,0)$, where units are metres. Cables are laid in straight lines and the widths of cables can be neglected.

An existing cable $C$ starts at the main switching site and goes in the direction $(3,1,-2)$.

A new cable is installed which passes through points $P(1,2,-1)$ and $Q(5,7,a)$.

(i) Find the value of $a$ for which $C$ and the new cable will meet.

To ensure that the cables do not meet, the engineers use $a=-3$. The engineers wish to connect each of the points $P$ and $Q$ to a point $R$ on $C$.

(ii) The engineers wish to reduce the length of cable required and believe in order to do this that angle $PRQ$ should be $90^\circ$. Show that this is not possible.

(iii) The engineers discover that the ground between $P$ and $R$ is difficult to drill through and now decide to make the length of $PR$ as small as possible. Find the coordinates of $R$ in this case and the exact minimum length.

The line L has equation $\mathbf r=(1,3,7)+t(2,-1,5)$, $t\in\mathbb R$. A,B have position vectors $(9,3,26)$ and $(13,9,\alpha)$. L intersects the line through A,B at P. Find α and the acute angle between L and AB. [7]

Line l passes through A=(1,2,−1) and B=(2,a,1).

(i) Show that l cuts the x-axis when a=−2. [3]

(ii) Find the length of projection of OA onto l and hence the shortest distance from the origin to l. [5]