Find the point at which the line ⟨−2,−2,0⟩+𝑡⟨−3,2,−1⟩ ⟨ − 2 , − 2 , 0 ⟩ + t ⟨ − 3 , 2 , − 1 ⟩ intersects the plane −3𝑥+2𝑦−2𝑧=62 − 3 x + 2 y − 2 z = 62 .

Question

Find the point at which the line ⟨−2,−2,0⟩+𝑡⟨−3,2,−1⟩ ⟨ − 2 , − 2 , 0 ⟩ + t ⟨ − 3 , 2 , − 1 ⟩ intersects the plane −3𝑥+2𝑦−2𝑧=62 − 3 x + 2 y − 2 z = 62 .

Answer 1

Parametric equation of given line: x = - 2 - 3t, y = - 2 + 2t, z = - t. Substitute in equation of plane:

-3(- 2 - 3t) + 2(- 2 + 2t) - 2(- t) = 62; 6 + 9t - 4 + 4t + 2t = 62; 15t = 60; t = 4

x = - 2 - 3·4 = - 14, y = - 2 + 2·4 = 6; z = - 4

(- 14, 6, - 4) is point of intersection.