Find the distance between the parallel line and the plane.
line: vector r = [2,0,1]+t[1,4,1]
plane: 2x-y+2z=4
To find the distance between a parallel line and a plane, we can use the formula for the distance between a point and a plane.
In this case, we need to find the distance between the line and the plane. The line is given by the vector equation r = [2,0,1] + t[1,4,1], where t is a parameter.
To find the distance, we need to find a point on the line and calculate the perpendicular distance from that point to the plane.
Step 1: Find a point on the line:
Let's take t = 0 and substitute it into the equation r = [2,0,1] + t[1,4,1]:
r = [2,0,1] + 0[1,4,1] = [2,0,1]
So, a point on the line is P = [2,0,1].
Step 2: Calculate the distance from the point P to the plane:
The distance d from a point (x0, y0, z0) to a plane Ax + By + Cz + D = 0 is given by the formula:
d = |Ax0 + By0 + Cz0 + D| / sqrt(A^2 + B^2 + C^2)
In this case, the equation of the plane is 2x - y + 2z = 4, so A = 2, B = -1, C = 2, and D = -4.
Substituting the values of P = [2,0,1] along with the coefficients of the plane into the formula, we get:
d = |2*2 - (-1)*0 + 2*1 - 4| / sqrt(2^2 + (-1)^2 + 2^2)
= |4 + 0 + 2 - 4| / sqrt(4 + 1 + 4)
= |2| / sqrt(9)
= 2 / 3
Therefore, the distance between the parallel line and the plane is 2/3 units.