This exercise explores key relationships between a pair of lines. Consider the f

Question

This exercise explores key relationships between a pair of lines. Consider the following two lines: one with parametric equations x(s)=4-2s, y(s) = 2 + s, z(s)=1+3s, and the other being the line through (-4, 2, 17) in the direction v = (-2, 1, 5).

(a) Find a direction vector for the first line, which is given in parametric form.

(b) Find parametric equations for the second line, written in terms of the parameter t.

(c) Show that the two lines intersect at a single point by finding the values of s and t that result in the same point.

(d) Find the angle formed where the two lines intersect, noting that this angle will be given by the angle between their respective direction vectors.

(e) Find an equation for the plane that contains both of the lines described in this problem

Explanation / Answer

a) x = 4 -2s ; y = 2 +s ; z = 1 +3s

direction vector : ( -2 , 1 ,3 )

b) Parametric equation : (-4, 2, 17) in the direction v = (-2, 1, 5)

x = -4 + -2t ; y = 2 +t ; z = 17 + 5t

c) Point of intersection of two lines :

-4 -2s = 4 -2t -----(1)

2 +s = 2 +t ----(2)

s = t ;

1 +3s = 17 +5t

1 +3s = 17 +5s ----> 2s = -16 ; s = -8 ; t = -8

Point they intersect is ( 4 -2*-8 , 2 -8 , 1 +3*-8) = ( 20 , -6 , -23)

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