To solve this problem, we'll go step by step.
a) To find the coordinates of a point on the line, we need to choose a value for one of the variables and solve for the other two. Let's choose a value for z, for example, z = 0. Plugging z = 0 into the equation, we have:
x - 1/3 = y + 2/4 = 0 - 4/5
Simplifying these equations, we get:
x - 1/3 = y + 1/2 = -4/5
To find the values of x and y, we can use any two of the equations above. Let's choose the first two equations:
x - 1/3 = y + 1/2
Rearranging this equation, we have:
x = y + 1/2 + 1/3
x = y + 5/6
So, the coordinates of the point on the line are (y + 5/6, y, 0).
b) To find the vector parallel to the line, we can use the coefficients of x, y, and z in the given equation. The coefficients of x, y, and z are 1, -1, and 1/5, respectively. Therefore, the vector parallel to the line is (1, -1, 1/5).
c) To write down the equation of the line in parametric form, we can express x, y, and z in terms of a parameter t. Using the coordinates of the point on the line (y + 5/6, y, 0) from part (a), we have:
x = y + 5/6
y = y
z = 0
So, the parametric equations of the line are:
x = t + 5/6
y = t
z = 0
d) To determine the Cartesian equation of the plane perpendicular to the line and passing through the point (2, 3, -1), we need the normal vector of the plane. The normal vector is the coefficients of x, y, and z in the line's equation, which are 1, -1, and 1/5, respectively.
Using the point-normal form of a plane equation, the equation of the plane is:
1(x - 2) + (-1)(y - 3) + (1/5)(z + 1) = 0
Expanding and simplifying this equation, we get:
x - 2 - y + 3 + (1/5)z + 1/5 = 0
x - y + (1/5)z + 10/5 = 0
5x - 5y + z + 10 = 0
Therefore, the Cartesian equation of the plane is 5x - 5y + z + 10 = 0.