Determine whether the lines intersect, and if so, find the point of intersection. (If an answer does not exist, enter DNE.)

Question

x = 6t + 2,    y = 7,    z = −t + 1

x = 2s + 2,    y = 2s + 7,    z = s + 1

(x, y, z) = ( ___ )

If the lines intersect, find and the angle between the lines. (Round your answer to one decimal place. If an answer does not exist, enter DNE.)

𝜃 = ____ °

Answer 1

6t + 2 = 2s + 2; 7 = 2s + 7; - t + 1 = s + 1

6t = 2s; 2s = 0 - t = s;

s = 0, t = 0.

So, (x, y, z) = (2, 7, 1) iz intersection point

Dirction vector of 1st line u = <6, 0, -1>; direction vector of 2nd line: v = <2, 2, 1>

cosθ = (6·2 + 0·2 - 1·1>/(√37·√9) = 11/(3√37); θ = cos^-1(11/(3√37) = 52.9°