Question

determine whether the line of parametric equations, x = 5+3t, y = −2t, z = 1+t, t is in ℝ, intersects the plane with equation 3x+4y+6z−7 = 0.

Answer 1

To determine if the line of parametric equations intersects the plane, substitute the values of x, y, and z into the equation of the plane and solve for t.

Step-by-step explanation:

To determine if the line of parametric equations intersects the plane, we need to find the values of t for which the parametric equations satisfy the equation of the plane.

Substituting the values of x, y, and z from the parametric equations into the equation of the plane, we get: 3(5+3t) + 4(-2t) + 6(1+t) - 7 = 0.

Simplifying this equation gives us 15t + 8t + 6t - 6 = 0.

Combining like terms, we have 29t = 6.

Dividing both sides by 29, we find that t ≈ 0.207.

Since t is a real number, the line of parametric equations does intersect the plane.