Review practice · Section 1 of 3

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 marks

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

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

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

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 marks

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

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 marks

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

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 marks

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 marks

(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 marks

(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 marks

Relative to O, A and B have position vectors $2\mathbf i+\mathbf j+2\mathbf k$ and $3\mathbf i-2\mathbf j+p\mathbf k$, where p is a constant.

(a) Find p for which OA and OB are perpendicular.2 marks

(b) In the case p=6, find angle AOB to the nearest degree.3 marks

(c) In the case p=5, find the position vector of M on segment AB such that AM:MB=1:2.2 marks

(d) Find the values of p for which the length of AB is $\sqrt{11}$ units.4 marks

Relative to O, A,B,C have position vectors $7\mathbf i+6\mathbf j-\mathbf k$, $-11\mathbf i-3\mathbf j+26\mathbf k$, and $4\mathbf i+\mathbf j+7\mathbf k$. P lies on BA produced such that 3AB=PB.

(i) Find the position vector of P.2 marks

(ii) Find angle ABC.3 marks

(iii) Find the exact area of triangle ABC.3 marks