Review practice · Section 3 of 3

Relative to O, A,B have position vectors a,b and c is the position vector of C on AB dividing AB in the ratio 3:1. Given angle AOB is acute, show that the length d of the projection of OC onto OB is $d=\frac34|\mathbf b|+\frac{\mathbf a\cdot\mathbf b}{4|\mathbf b|}$.

Referred to O, $OA=\mathbf a$, $OB=\mathbf b$. C lies on OA with OC:CA=1:2; D lies on OB with OD:DB=2:3. M is the midpoint of CD.

(i) Find OM in terms of a,b and show that the area of triangle OMD is $k|\mathbf a\times\mathbf b|$, finding k.5 marks

(ii) Now $\mathbf a=2\mathbf i+3\mathbf j+6\mathbf k$, $\mathbf b=10\mathbf i+5\mathbf j+10\mathbf k$. Find $\overrightarrow{OM}\times\widehat{\mathbf b}$ and give the geometrical meaning of its magnitude.3 marks