Given M1(1,-2,3), M2(-3,5,-2) write the vector form and parametric form of the equation of the line M1M2.

We'll write the vector form of the equation of the line
M1M2.

r = r1 + t(r2 - r1)
(*),

where r1 and r2 are the vectors of position of the
points M1 and M2.

r1 = xM1*i + yM1*j +
zM1*k

We'll substitute the coordinates of
M1:

r1 = 1*i + (-2)*j + 3*k

r1
= i - 2j + 3k (1)

We'll write the equation of the vector
r2:

r2 = xM2*i + yM2*j +
zM2*k

r2 =  -3*i + 5*j +
(-2)*k

r2 =  -3i + 5j - 2k
(2)

We'll compute the
difference:

r2 - r1 = (-3-1)i + (5+2)j +
(-2-3)k

r2 - r1 = -4i + 7j - 5k
(3)

We'll substitute (1), (2), (3) in
(*):

r = i - 2j + 3k + t( -4i + 7j -
5k)

The vector form of the equation of the line that passes
through M1 and M2 is:

r = i - 2j + 3k +
t( -4i + 7j - 5k)

Knowing the vector form of
the equation, we'll write the parametric
form:

x = 1 -
4t

y = -2 +
7t

z = 3 -
5t