Review practice · Section 2 of 3

Referred to O, $OA=\mathbf a$, $OB=\mathbf b$, with OA perpendicular to OB. C,D lie on AB such that $AC:CB=\lambda:(1-\lambda)$ and $AD:DB=\mu:(1-\mu)$, where $0<\lambda<\mu<1$.

(i) Find OC in terms of a,b,λ and OD in terms of a,b,μ. By considering their vector product, show that the area of triangle OCD is $\frac12(\mu-\lambda)|\mathbf a\times\mathbf b|$.5 marks

Now $\mathbf a=2\mathbf i+3\mathbf j-\mathbf k$, $\mathbf b=-\mathbf i+2\mathbf j+4\mathbf k$ and AC:CD:DB=2:3:4.

(ii) Use (i) to calculate the exact area of triangle OCD.3 marks

Referred to O, A,B have position vectors $\mathbf a=\mathbf i-\mathbf j+\mathbf k$, $\mathbf b=2\mathbf i+3\mathbf k$. C has position vector $\mathbf c=\lambda\mathbf a+\mu\mathbf b$, where λ,μ are constants.

(i) Find the exact area of triangle OAB.3 marks

(ii) Given that OABC forms a parallelogram, write down λ and μ.2 marks

(iii) Given instead μ=2 and $OC=\sqrt{59}$, find the possible coordinates of C in exact form.4 marks

Referred to O, A,B have position vectors $\mathbf a=\mathbf i-\mathbf j+\mathbf k$, $\mathbf b=\mathbf i+2\mathbf j$. C has position vector $\mathbf c=\lambda\mathbf a+\mu\mathbf b$, where λ and μ are positive constants.

(i) Given the area of OAC is $\sqrt{126}$, find μ.4 marks

(ii) Given instead μ=4 and $OC=5\sqrt3$, find the possible coordinates of C.4 marks

Given $\mathbf n=(\mathbf b\cdot\mathbf b)\mathbf a-(\mathbf a\cdot\mathbf b)\mathbf b$, show that n is perpendicular to b, if a,b are two non-zero vectors.

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 marks

(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 marks

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

(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 marks

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 marks

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

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

(iv) Given a,b are unit vectors and angle AOB=60°, find the length of the projection of OC onto OA.3 marks