Question

Determine the equation of the line that is perpendicular to the lines r(t)=(-2+3t,2t,3t)

Answer 1

Vector Equation
(Line)(x,y) = (x,y) + t(a,b);tERParametric Formx = x + t(a), y = y + t(b); tERr = (-4,-2) + t((-3,5);tERFind the vector equation of the line passing through A(-4,-2) & parallel to m = (-3,5)Point: (2,5)
Create a direction vector: AB = (-1 - 2, 4 - 5) 
= (-3,-1) or (3,1)when -1 (or any scalar multiple) is divided out.
r = (2,5) + t(-3,-1);tERFind the vector equation of the line passing through A(2,5) & B(-1,4)x = 4 - 3t
y = -2 + 5t
;tERWrite the parametric equations of the line passing through the line passing through the point A(4,-2) & with a direction vector of m =(-3,5)Create Vector Equation first:
AB = (2,8)
Point: (4,-3)
r = (4,-3) + (2,8); tER
x = 4 + 2t 
y = -3 + 8t
;tERWrite the parametric equations of the line through A(4,-3) & B(6,5)Make parametric equations: 
x = 5 + 4t
y = -2 + 3t ; tER
For x sub in -3
-3 = 5 + 4t 
(-8 - 5)/4 = t
-2 = t
For y sub in -8 
-8 = -2 + 3t
(-8 + 2)/3 = t
-2 = t
Parameter 't' is consistent so pt(-3,-8) is on the line.Given the equation r = (5,-2) + t(4,3);tER, is (-3,-8) on the line?Make parametric equations: 
x = 5 + 4t
y = -2 + 3t ; tER
For x sub in 1
-1 = 5 + 4t 
(-1 - 5)/4 = t
-1 = t
For y sub in -7 
-7 = -2 + 3t
(-7 + 2)/3 = t
-5/3 = t
Parameter 't' is inconsistent so pt(1,-7) is not on the line.Given the equation r = (5,-2) + t(4,3);tER, is (1,-7) on the line?Use parametric equations when generating points: 
x = 5 + 4t
y = -2 + 3t ;tER
X-int:
sub in y = 0
0 = -2 + 3t
solve for t
2/3 = t (this is the parameter that will generate the x-int) 
Sub t = 2/3 into x = 5 + 4t 
x = 5 + 4(2/3)
x = 5 + (8/3)
x = 15/3 + (8/3) 
x = 23/3
The x-int is (23/3, 0)What is the x-int of the line r = (5,-2) + t(4,3); tER?Note: if they define the same line: 1) Are their direction vectors scalar multiples? 2) Check the point of one equation in the other equation (LS = RS if point is subbed in)What are the two requirements for 2 lines to define the same line?