Practice section 9 of 9

OABC is the base of a pyramid where O is the origin. E is the highest point and D lies on OB with DE perpendicular to the base. The position vectors of A,B,C,E are respectively $-7\mathbf j+\mathbf k$, $3\mathbf i-9\mathbf j+3\mathbf k$, $4\mathbf i-5\mathbf j+3\mathbf k$, and $6\mathbf i-7\mathbf j-5\mathbf k$.

(i) Find angle AOE. [2]

(ii) Find OD, the length of the projection of OE onto OB. Hence or otherwise show that D has position vector $2\mathbf i-6\mathbf j+2\mathbf k$. [4]

(iii) Show A,D,C are collinear. [2]

(iv) Show OB is perpendicular to AC. Hence find the exact area of OABC in the form $m\sqrt{66}$, where m is a constant to be determined. [3]

Referred to O, $OA=\mathbf a$, $OB=\mathbf b$. C is on AB produced such that 2AB=3BC.

(i) Find OC in terms of a,b. [2]

(ii) Show that the area of triangle OBC can be written as $k|\mathbf a\times\mathbf b|$, finding k. [2]

(iii) D is on OB such that area (OAD):area (OBC)=4:3. Find OD in terms of b. [3]

(iv) Given a,b are unit vectors and angle AOB=60°, find the length of the projection of OC onto OA. [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]

(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]