Question

Find a vector equation and parametric equations for the line. (Use the parameter t.)

The line through the point (0, 13, −10) and parallel to the line x = −1 + 2t, y = 6 − 2t, z = 3 + 9t
r(t)=(x(t), y(t), z(t)) =______

Answer 1

To find the vector and parametric equations for a line through a point that is parallel to another line, use the direction vector of the given line and the point of application. The vector equation is r(t) = (0, 13, -10) + t(2, -2, 9), and the parametric equations are x(t) = 2t, y(t) = 13 - 2t, z(t) = -10 + 9t.

Step-by-step explanation:

To find the vector equation and parametric equations for a line that passes through a given point and is parallel to another line, you can follow these steps:

• Determine the direction vector of the given parallel line.
• Use the given point as the point of application for the new line.
• Write the vector equation and parametric equations using the point and direction vector.

The line provided, x = -1 + 2t, y = 6 - 2t, z = 3 + 9t, has a direction vector (2, -2, 9). This is obtained by taking the coefficients of t in each of the parametric equations. Since the new line must be parallel to the given line, it will have the same direction vector.

The point through which the new line passes is (0, 13, -10).

The vector equation of the line is:

r(t) = (0, 13, -10) + t(2, -2, 9)

The parametric equations are:

x(t) = 0 + 2t = 2t

y(t) = 13 - 2t

z(t) = -10 + 9t