Practice section 2 of 6

The points A,B,P have position vectors $\mathbf i+2\mathbf j+3\mathbf k$, $2\mathbf i+4\mathbf j+4\mathbf k$ and $\mathbf i+\mathbf j-\mathbf k$, respectively. Find the perpendicular distance from P to the line through A and B.

The diagram depicts a cuboid with vertices A,B,C,D,E,F,G,H. Write down whether each pair of lines is parallel, intersecting or skew: AC and CD; AE and DH; AE and CD; AF and BE; DG and BE; AD and GF; AH and CF; AH and BE.

For each pair, test whether the lines are parallel, intersecting or skew. If they intersect, find the position vector of their intersection. All parameters are real.

(a) $l_1:\mathbf r=(1,2,3)+\lambda(1,2,1)$ and $l_2:\mathbf r=(1,3,4)+\mu(2,4,2)$.

(b) $l_1:\mathbf r=(1,2,3)+\lambda(1,2,1)$ and $l_2:\mathbf r=(0,1,5)+\mu(1,3,4)$.

(c) $l_1:\mathbf r=(1,2,3)+\lambda(1,2,1)$ and $l_2:\mathbf r=(0,1,2)+\mu(1,3,4)$.

Find the acute angle between $l_1:\mathbf r=(4,3,1)+\lambda(1,2,1)$ and $l_2:\mathbf r=(-1,0,-1)+\mu(3,2,6)$, with λ,μ real.

Given $l_1:\mathbf r=(1,2,3)+\lambda(1,2,5)$ and $l_2:\mathbf r=(2,1,7)+\mu(1,5,6)$, with λ,μ real, find (i) their point of intersection and (ii) their acute angle.

With respect to origin O, A and C have position vectors $3\mathbf i+11\mathbf j+11\mathbf k$ and $2\mathbf i-\mathbf j+10\mathbf k$, respectively. B is such that OABC is a parallelogram.

(i) Write down the position vector of B.

(ii) Find angle OAB to the nearest degree.

(iii) Find a vector equation of line AB.

D has position vector $-\mathbf i+13\mathbf j-9\mathbf k$.

(iv) Verify that D is on line AB.

(v) Find the position vector of the point on line AB closest to O.