Review practice · Section 2 of 2

Line $l_1$ passes through A=(−1,−2,1), B=(0,1,5). Line $l_2$ has equation $x-1=(y-2)/2=z-3$. They intersect at A. Find:

(i) vector equations of $l_1,l_2$;2 marks

(ii) their acute angle;2 marks

(iii) the position vector of the perpendicular foot F from B onto $l_2$;3 marks

(iv) the equation of the line $l_3$, the mirror image of $l_1$ in $l_2$.3 marks

With respect to O, A,B,C have position vectors $\mathbf a=(1,\sin t,0)$, $\mathbf b=(\cos t,2,1)$, $\mathbf c=(2,-1,1)$, where $0\le t<\pi$.

(a) Find the exact t given a is perpendicular to b.3 marks

(b) X is on AB produced with AB:BX=1:4, and ACXY is a parallelogram. Take $t=\pi/2$ for the rest of the question.

(i) Find the position vectors of X,Y.4 marks

(ii) Find the area of ACXY. Hence find the shortest distance from X to line AC.3 marks

(iii) Line $l_1$ has equation $(x-2)/6=(3-y)/h$, z=k, where h,k are constants. The line through A,B intersects $l_1$ at a right angle. Find h,k.4 marks

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 marks

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

(iii) Find the area of OBAC.2 marks

(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 marks

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 mark

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 marks

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 mark

Paul stands at C=(−7,5,2).

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

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 mark

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

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