Direction vector

The line p is given by the point P [- 0,5; 1] and the direction vector s = (1,5; - 3) determines:
A) value of parameter t for points X [- 1,5; 3], Y [1; - 2] lines p
B) whether the points R [0,5; - 1], S [1,5; 3] lie on the line p
C) parametric equations of the line m || p if the line m passes through the origin of the coordinate system
D) parametric equations of the line n _ | _ p, if the line n passes through the point N [- 2; 4]

Correct answer:

t1 =  -0.6667
t2 =  1
r =  1
S =  0
mx = 1.5t
my = -3t
nx = -2+3t
ny = 4-1.5t

Step-by-step explanation:

P=(−0.5,1) s=(1.5,−3)  X [ − 1,5;3] −1.5 = Px​ + t1​⋅ sx​  t1​=(−1.5−Px​)/sx​=(−1.5−(−0.5))/1.5=−0.6667
Y [1; − 2] 1 = Px​ + t1​⋅ sx​  t2​=(1−Px​)/sx​=(1−(−0.5))/1.5=1
R=(0.5,−1)  p1​=(Rx​−Px​)/sx​=(0.5−(−0.5))/1.5=32​≐0.6667 p2​=(Ry​−Py​)/sy​=((−1)−1)/(−3)=32​≐0.6667 p1​=p2​ r=1
  S [1,5;3] p3​=(1.5−Px​)/sx​=(1.5−(−0.5))/1.5=34​=131​≐1.3333 p4​=(3−Py​)/sy​=(3−1)/(−3)=−32​≐−0.6667 p3​ ​= p4​ S=0
mx=mx​=1.5t
my=my​=−3t
(1.5; −3) ⊥ (3; 1.5) nx=nx​=−2+3t
ny=ny​=4−1.5t

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