Prove that the line d is parallel to the plane P if the equation of the plane is 2x+3y+z-1=0.The parametric equations of the line are x=1+2t,...

The line d is parallel to the plane P, if and only if the
vector parallel to the line, v, and the vector perpendicular to the plane P, n, are
perpendicular.

Two vectors are perpendicular if and only if
their dor product is zero:

v*n =
0

Since the equation of the plane P is given, we'll
identify the parametric coefficients of the normal vector to the plane,
n.

2x+3y+z-1=0, where n(2 , 3 ,
1)

We'll write n = 2i + 3j +
k

Since the position vector of the line d
is:

r = r0 + t*v

We'll write
the parametric equations of d:

x = x0 +
t*vx

y = y0 + t*vy

z = z0 +
t*vz

Comparing the given parametric equations and the
general parametric equations, we'll identify the parametric coefficients of the vector
v:

x=1+2t

y=2-3t

z=5t

v
(2 , -3 , 5)

v = 2i - 3j +
5k

Now, we'll write the dot product of n and
v:

n*v = 2*2 + 3*(-3) + 1*5

If
n*v = 0, the line d is parallel to the plane P.

n*v = 4 - 9
+ 5

n*v =
0

Since n*v = 0, the given line is parallel
to the plane P.