Determine an equation for the line parallel to 2x + 6y + 4z = 1 and contains the point P(3, 2, 1).
To find an equation for the line parallel to 2x + 6y + 4z = 1, we need to determine the direction of the line. We can do this by looking at the coefficients of x, y, and z in the given equation.
The coefficients of x, y, and z in the given equation are 2, 6, and 4, respectively. These coefficients represent the direction ratios of the line. The direction ratios indicate how much the x, y, and z coordinates change for every unit increase in distance along the line.
So, the direction ratios of the line parallel to 2x + 6y + 4z = 1 are 2, 6, and 4.
Now, we can use these direction ratios and the point P(3, 2, 1) to find the equation of the line using the point-normal form of a line equation.
The point-normal form of a line equation is given by:
(x - x₁) / a = (y - y₁) / b = (z - z₁) / c
Here, (x₁, y₁, z₁) represents the coordinates of the given point P(3, 2, 1), and (a, b, c) represents the direction ratios of the line.
Substituting the values, we get:
(x - 3) / 2 = (y - 2) / 6 = (z - 1) / 4
Therefore, an equation for the line parallel to 2x + 6y + 4z = 1 and containing the point P(3, 2, 1) is:
(x - 3) / 2 = (y - 2) / 6 = (z - 1) / 4