Practice section 6 of 6

Relative to O, A,B,C have position vectors $\mathbf a=3\mathbf i-\mathbf j$, $\mathbf b=3\mathbf i+4\mathbf j-12\mathbf k$, $\mathbf c=\lambda\mathbf j+12\mathbf k$. Line $l_1$ through A,C has equation $(x-3)/3=(y+1)/4=-z/12$.

(i) Show λ=−5. [2]

(ii) Find the position vector of P on OA with BP perpendicular to OA. Hence or otherwise prove OBAC is a rhombus. [5]

(iii) Find the area of OBAC. [2]

(iv) Line $l_2$ passes through O,Q, where Q lies on BA produced with BA/BQ=1/4. Find the acute angle between $l_1,l_2$. [3]

Referred to O, A,B,C have position vectors a,b,c. C lies on AB produced with AC:AB=k:1, where 1<k<2. Show $\mathbf c=(1-k)\mathbf a+k\mathbf b$. [1]

D is the midpoint of OA. Lines DB produced and OC produced meet at E. Given k=3/2, find the position vector of E in terms of a,b and the ratio DB:DE. [4]

Distances are in metres and time in seconds. A radio-controlled airplane takes off from ground level and travels at a steady speed in a straight line. Its position t seconds after takeoff is $\mathbf r=\mathbf a+t\mathbf b$, where a is its departure position and b its velocity vector.

(i) After 6 seconds it reaches P=(−9,4,6) and its velocity is $-2\mathbf i+\mathbf j+\mathbf k$. Find its departure coordinates. [1]

Paul stands at C=(−7,5,2).

(ii) Find the shortest distance from Paul to the airplane’s flight path. [3]

A drone takes off at the same time and flies at steady speed on the straight line $(2x-1)/(-6)=(y+7)/4=(z-10)/k$.

(iii) Show its flight path can be written $\mathbf r=\mathbf p+\lambda\mathbf q$, λ≥0, determining p,q. [1]

(iv) Given the flight paths intersect, find k. [3]

(v) Do the airplane and drone collide? Justify your answer. [1]