1 a Find the parametric and symmetric equations of the line

1. a) Find the parametric and symmetric equations of the line l that passes through the point A(6,4,0) and is parallel to the line passing though B(-2,0,4) and C(3,-2,1).

b) If (-4,m,n) is a point on the line l, find the values of m and n?

2. For what value(s) of s is the vector (6,3,s) a linear combination of the vectors (1,2,3) and (0,1,2)?

I`m stuggling these little questions, if could help me out that would be great. Thank you in advance!

Solution

a) Parametric form : L = P + tv

where P = (6, 4, 0) and v would be th evector passing through(-2,0,4) and C(3,-2,1)

v = ( 5, -2 , -3)

L = (6, 4, 0) +t(3, -2, 1)

x = 2+3t ; y = 4 -2t ; z= t

Symmeteric form : ( x-2)/3 = (y -4)/-2 = z

b) If point (-4, m , n) is on line then:

x= 2+3t ---> -4 = 2+3t

t = -2

So, m = y = 4-2t = 4 +4 = 8

n = z = t = -2