Question

find parametric equations for the line through (2, 2, 8) that is perpendicular to the plane x − y 4z = 2. (use the parameter t.) (x(t), y(t), z(t)) =

Answer 1

To find the parametric equations for the line through (2, 2, 8) that is perpendicular to the plane x − y − 4z = 2, you can follow these steps: find the normal vector to the plane, consider the vector form of the line equation, find the direction vector of the line, and finally write the parametric equations.

Step-by-step explanation:

To find the parametric equations for the line through (2, 2, 8) that is perpendicular to the plane x - y - 4z = 2, follow these steps:

1. Find the normal vector to the plane by extracting the coefficients of x, y, and z from the plane equation. In this case, the normal vector is (-1, 1, -4).
2. Consider the vector form of the line equation, which is given as [x(t), y(t), z(t)] = [2, 2, 8] + t * (a, b, c), where (a, b, c) is the direction vector of the line.
3. Since the line is perpendicular to the plane, the direction vector (a, b, c) must be orthogonal to the normal vector of the plane. Therefore, take the dot product of (a, b, c) and the normal vector (-1, 1, -4) and set it equal to zero to find the direction vector.
4. Solve the dot product equation to obtain the direction vector (a, b, c). In this case, the direction vector is (4, 4, 1).
5. Write the parametric equations for the line by substituting the values into the equation: x(t) = 2 + 4t, y(t) = 2 + 4t, z(t) = 8 + t.