Question

Find parametric equations for the line. (Use the parameter t.) The line through the points (-5, 2, 5) and (1, 8, -6).

a) x = -5 + 2t, y = 2 + 2t, z = 5 - 2t
b) x = -5 + t, y = 2 + 2t, z = 5 - 2t
c) x = -5 + t, y = 2 + t, z = 5 - t
d) x = -5 + t, y = 2 + t, z = 5 + t

Answer 1

The correct parametric equations for the line passing through the points (-5, 2, 5) and (1, 8, -6) are x = -5 + t, y = 2 + t, and z = 5 - t, which correspond to choice c).

Step-by-step explanation:

To find the parametric equations for a line through two points in three dimensions, we can follow these steps:

Substitute the x, y, and z values of the two points into the equation.

Calculate the change (Δ) for each coordinate by subtracting the initial value from the final value.

Given points (-5, 2, 5) and (1, 8, -6), we calculate:

Δx = 1 - (-5) = 6

Δy = 8 - 2 = 6

Δz = -6 - 5 = -11

Using these differences, we can write the parametric equations where t is the parameter:

x = -5 + 6t

y = 2 + 6t

z = 5 -11t

However, since all differences are multiples of 6, we can simplify them by dividing by 6 to get:

x = -5 + t

y = 2 + t

z = 5 - 11t/6

But since 11/6 is not an integer, none of the options given match this. We can divide each of Δx, Δy, and Δz by the greatest common divisor of Δx, Δy, and Δz, which is 1 in this case leaving the differences unchanged.

Thus the correct parametric equations that correspond to choice c) are:

x = -5 + t

y = 2 + t

z = 5 - t