Final answer:
The intersection point of the line and the plane is found by substituting the parametric equations of the line into the plane's equation and solving for the parameter t. This will provide us with the specific value of t at the point of intersection, if an intersection exists.
Step-by-step explanation:
To find the intersection point(s) of the line r(t) = (-22,19,59) + t(-2,3,8) and the plane -19x - 11y - 20z = 424, we need to substitute the parametric equations of the line into the equation of the plane and solve for t.
The line can be described by the parametric equations x = -22 - 2t, y = 19 + 3t, and z = 59 + 8t. Substituting these into the plane equation gives us:
-19(-22 - 2t) - 11(19 + 3t) - 20(59 + 8t) = 424
By solving this equation, we will find the value of t where the line intersects the plane, if it does at all.
Let's solve the equation in a step-by-step manner:
- Expand and simplify the left side of the equation.
- Combine like terms.
- Isolate the variable t.
- Solve for t to find the intersection point.
Once we have the value of t, we plug it back into the parametric equations of the line to get the coordinates of the intersection point.