Practice section 8 of 9

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]

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

(c) In the case p=5, find the position vector of M on segment AB such that AM:MB=1:2. [2]

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

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]

(ii) Find angle ABC. [3]

(iii) Find the exact area of triangle ABC. [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]

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]

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]

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

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

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]

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

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.