4 Example 1Find the parametric and symmetric equations of the line L that passes through (1,-2,4) and is parallel to v = <2,4,-4>
5 Example 1 solutionFind parametric and symmetric equations of line L that passesthrough the point (1,-2,4 ) and is parallel to v = <2,4,-4>Solution to find a set of parametric equations of the line, use the coordinates x1 =1, y1=-2, z1 = 4 and direction numbers a=2, b = 4 and c=-4x= 1+2t, y = -2+4t, z=4-4t (parametric equations)Because a,b and c are all nonzero, a set of symmetric equations is
6 Example 2Find a set of parametric equations of the line that passes through the points (-2,1,0) and (1,3,5), Graph the equation (next slide shows axis).
9 Problem 26 Determine if any of the following lines are parallel. L1: (x-8)/4 = (y-5)/-2 = (z+9)/3L2: (x+7)/2 = (y-4)/1 = (z+6)/5L3: (x+4)/-8 = (y-1)/4 = (z+18)/-6L4: (x-2)/-2 = (y+3)/1 = (z-4)/1.5
10 Problem 28Determine whether lines intersect, and if so find the point of intersection and the angle of intersection.x = -3t+1, y = 4t + 1, z = 2t+4x=3s +1, y = 4s +1, z = -s +1
11 Solution 28 x = -3t+1, y = 4t + 1, z = 2t+4 x=3s +1, y = 4s +1, z = -s +1Set x y and z equations equal-3t+1 =3s s +1 = 4t s +1 = 2t+4From the first one we get s=-tWhen s=-t is plugged into the second we get t=1/3When plugged into the third equation we get t = -3Hence the lines do not intersect
12 Problem 30Determine whether lines intersect, and if so find the point of intersection and the angle of intersection.(x-2)/-3 = (y-2)/6 = z-3(x-3)/2=y+5=(z+2)/4
13 Hint on Problem 30 (x-2)/-3 = (y-2)/6 = z-3 (x-3)/2=y+5=(z+2)/4 Write the equations in parametric formThen solve as per #28 These do intersect.Use the dot product of the direction vectors to find the angle between the two lines<-3,6,1> ∙ <2,1,4>