Two laser guidance beams used in a large-scale precision engineering project are modelled as lines l1 l_1\,l1 and l2 l_2\,l2 relative to a fixed origin OOO. The equations of the lines are given by:
l1:r=(i+kj)+t(i−2j+2k) l_1: \mathbf{r} = (\mathbf{i} + k\mathbf{j}) + t(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) l1:r=(i+kj)+t(i−2j+2k) l2:r=(i+4j+6k)+s(2i+j−2k) l_2: \mathbf{r} = (\mathbf{i} + 4\mathbf{j} + 6\mathbf{k}) + s(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}) l2:r=(i+4j+6k)+s(2i+j−2k)where t t\,t and s s\,s are scalar parameters and k k\,k is a constant.
Given that the two laser beams intersect at a single point:
find the value of kkk.
find the position vector of the point of intersection.
Calculate the acute angle between l1 l_1\,l1 and l2l_2l2. Give your answer in degrees to one decimal place.
The point A A\,A lies on l1 l_1\,l1 with parameter t=4t = 4t=4. The point B B\,B lies on l2 l_2\,l2 such that the vector AB⃗\vec{AB}AB is perpendicular to l2l_2l2.
Find the coordinates of BBB.
