(Solved):   x = t 1) The lines and have equations y = 1+5t, t = R and r = (2,2,1)+ (1, ...

 

x = t 1) The lines ? and ? have equations y = ?1+5t, t = R and r = (2,2,1)+ ?(?1,1,3), ? = R respectively. The z = 2+3t lines intersect at point P. a) Find the coordinates of P. b) Point Q has coordinates (-1,5,10). Show that Q lies on l?. c) Find the distance PQ. d) Let be the acute angle between the lines at point P. Find the exact value of cose. e) Let d be the shortest (perpendicular) distance from Q to line ?. Using elementary trigonometry and the results from parts c) and d), find the exact value of d. 2) Linear systems can be applied to chemistry, particularly when balancing chemical reactions. The only real premise that you need to keep in mind is the law of conservation of matter; the total number of atoms of one particular element on the left side of the reaction must equal the total number of atoms of the same element on the right side of the reaction. Consider the following reaction: xN?O5 ?yNO?+z0? a) Briefly EXPLAIN how I arrived at the following linear system: T?: 2x=y=0 in the context of T?: 5x-2y-2z=0 the above problem. b) The linear system described in part a) has a line of intersection. Find the equation of this line of intersection in parametric and vector form. c) Find an integer solution to balance the above chemical equation. Note: This type of analysis is how computer programs balance chemical reactions. Form a system of equations and apply linear algebra tools to solve them. We used substitution and/or elimination (which are not necessarily the best tools). Matrices are more efficient at dealing with such problems. ??: x-y-z=-2 3) Determine whether the planes ?: 3x+y+z=14 have a common point of intersection, common line of 73: 3x-y+z=4 intersection or are coincidental. If none of the above applies, describe any interactions between the planes.