Practice section 7 of 9

The points $A$, $B$, $C$ and $D$ have position vectors $\mathbf i-2\mathbf j+5\mathbf k$, $\mathbf i+3\mathbf j$, $10\mathbf i+\mathbf j+2\mathbf k$ and $-2\mathbf i+4\mathbf j+5\mathbf k$ respectively, with respect to an origin $O$. The point $P$ on $AB$ is such that $AP:PB=\lambda:1-\lambda$ and the point $Q$ on $CD$ is such that $CQ:QD=\mu:1-\mu$. Find $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ in terms of $\lambda$ and $\mu$ respectively.

Given that $\overrightarrow{PQ}$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{CD}$,

(i) show that $\overrightarrow{PQ}=\mathbf i+2\mathbf j+2\mathbf k$,

(ii) find the area of the triangle $ABQ$.

Relative to the origin $O$, three distinct fixed points $A$, $B$ and $C$ have position vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$ respectively. It is known that $\mathbf b$ is a unit vector, $|\mathbf a|=3$, $|\mathbf c|=2$ and the angle $AOC$ is $60^\circ$.

(i) State the geometrical interpretation of $|\mathbf b\cdot\mathbf c|$. [1]

It is further given that $\mathbf a\times2\mathbf b=\mathbf b\times\mathbf c$.

(ii) Find the ratio of the area of triangle $AOB$ to the area of triangle $BOC$. [2]

(iii) Show that $2\mathbf a+\mathbf c=k\mathbf b$ where $k\in\mathbb R$, $k\ne0$.

By considering $(2\mathbf a+\mathbf c)\cdot(2\mathbf a+\mathbf c)$, find the exact values of $k$. [6]

The position vectors of A, B and C are $\mathbf a=\mathbf i-\mathbf j+2\mathbf k$, $\mathbf b=3\mathbf i+5\mathbf j-2\mathbf k$ and $\mathbf c=p\mathbf j+4\mathbf k$ respectively.

(i) Find the unit vector in the direction of $\overrightarrow{AB}$. [2]

(ii) Find the value of p such that A, B and C are collinear. [3]

(iii) If $p=-1$, find the position vector of D such that ABCD is a parallelogram. [2]

(iv) If $|\overrightarrow{AC}|=3$, find the possible values of p. [2]

Relatively to an origin O, the position vectors of A, B and P are $\mathbf a$, $\mathbf b$ and $\frac12(\mathbf a+\mathbf b)$ respectively, where a and b are non-zero and non-parallel vectors. The point Q lies on the line segment OP such that OQ:QP=2:1. The line AQ cuts OB at R.

(i) Find, in terms of a and b, $\overrightarrow{AQ}$. [2]

(ii) Show that $\overrightarrow{OR}=\frac12\mathbf b$. [3]

Referred to the origin O, the points A and B have position vectors a and b respectively such that $|\mathbf a|=5$ and $|\mathbf b|=3$.

(i) It is given that $|2\mathbf b-\mathbf a|=7$. Using a suitable scalar product, show that $\mathbf a\cdot\mathbf b=3$. [2]

(ii) Deduce the length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$. [1]

The point P lies on AB such that AP:PB=1:k, where k is a positive constant.

(iii) Write down the position vector of P in terms of k,a and b and show that the area of triangle OAP is $\dfrac{3\sqrt6}{k+1}$. [5]

With respect to O, A,B,C have position vectors $\mathbf a=5\mathbf i+4\mathbf j+10\mathbf k$, $\mathbf b=-4\mathbf i+4\mathbf j-2\mathbf k$, $\mathbf c=-5\mathbf i+9\mathbf j+5\mathbf k$.

(i) Find the position vector of D such that ABCD is a parallelogram. [2]

(ii) N divides AB in the ratio $\lambda:(1-\lambda)$. Show that $\overrightarrow{CN}=(10-9\lambda)\mathbf i-5\mathbf j+(5-12\lambda)\mathbf k$. Hence find the position vector of N given CN is perpendicular to AB. [4]

(iii) P has position vector $(-1+3t)\mathbf i+4\mathbf j+(2+4t)\mathbf k$, where $t\in\mathbb R$. Show that P is collinear with A and B for all real t. [3]