Find the point at which the line ⟨0,4,−4⟩+𝑡⟨4,5,−3⟩ ⟨ 0 , 4 , − 4 ⟩ + t ⟨ 4 , 5 , − 3 ⟩ intersects the plane −𝑥+𝑦+𝑧=−10 − x + y + z = − 10 .

Question

Find the point at which the line ⟨0,4,−4⟩+𝑡⟨4,5,−3⟩ ⟨ 0 , 4 , − 4 ⟩ + t ⟨ 4 , 5 , − 3 ⟩ intersects the plane −𝑥+𝑦+𝑧=−10 − x + y + z = − 10 .

Answer 1

Substitute the values of x, y, and z ftom the parametric equations of the line into the equation of the plane and solve for t.

x = 4t

y = 4 + 5t

z = -4 -3t

Then 4t + 4 + 5t -4 -3t = - 10 ⇒ 6t = -10 ⇒ t = -5/3

By substituting the value of t = -5/3 back to the parametric equations of the line

we are getting the coordinates of the point of interest.

x = -20/3

y = 13/3

z = 1