Find the intersections of the following lines, if there are any

Question

r=(-2,0,-3) + t(5,1,3)

r= (5,8,-6) + s(-1,2,-3)

Please include steps and answer

Answer 1

At intersection point r₁ = r2 or in coordinates we have 3 equations with 2 variables t and s:

- 2 + 5t = 5 - s, t = 8 + 2s, - 3 + 3t = - 6 - 3s. We solve first 2 together and will see if satisfied 3rd equation.

5t + s = 7

3t + 3s = - 3,

5t + s = 7

t + s = - 1

sabtructing this equations we get 4t = 8, t = 2

Then s = - 1 - t = - 1 - 2 = - 3. Now we check if t = 2 and s = - 3 sutisfied equation t - 2s = 8, we have 2 - 2(-3) = 8, So 8 = 8. Now we avaluate coordinates of point of intersection: x = - 2 + 5·2 = 8. y = 2, z = - 3 + 3·2 = 3;

(8, 2, 3) is point of intersection