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:
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.