To find the intersection between the surface and the line, we need to:

1. Express both the surface equation and the parametric equation of the line.
2. Set the parametric coordinates of the line into the surface equation to solve for the parameter.

Step 1: Surface Equation

The equation of the surface is: $-2x + y - z + 3 = 0$ This can be rewritten as: $-2x + y - z = -3 \quad \text{(Equation 1)}$

Step 2: Parametric Equation of the Line

The given line formula is expressed in vector form: $\mathbf{r}(n) = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k} + n(-\mathbf{i} + 4\mathbf{j} + \mathbf{k})$ This can be split into components:

• The point $P(2, 2, 1)$ represents the starting point of the line.
• The direction vector is $(-1, 4, 1)$, which means the line moves in this direction as $n$ changes.

Thus, the parametric equations of the line are: $x(n) = 2 - n$ $y(n) = 2 + 4n$ $z(n) = 1 + n$

Step 3: Substitute Line Parametric Equations into the Surface Equation

Substitute $x(n)$, $y(n)$, and $z(n)$ into the surface equation: $-2x + y - z = -3$ Substitute the parametric expressions: $-2(2 - n) + (2 + 4n) - (1 + n) = -3$ Simplify the equation: $-4 + 2n + 2 + 4n - 1 - n = -3$ $-3 + 5n = -3$ Solve for $n$: $5n = 0 \quad \Rightarrow \quad n = 0$

Step 4: Find the Coordinates of the Intersection

Now that we know $n = 0$, substitute this value back into the parametric equations of the line to find the intersection point: $x(0) = 2 - 0 = 2$ $y(0) = 2 + 4(0) = 2$ $z(0) = 1 + 0 = 1$

Thus, the intersection point is: $(2, 2, 1)$

Conclusion

The intersection between the surface $-2x + y - z + 3 = 0$ and the line $2\mathbf{i} + 2\mathbf{j} + \mathbf{k} + n(-\mathbf{i} + 4\mathbf{j} + \mathbf{k})$ occurs at the point $(2, 2, 1)$.

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Would you like any further details or have any questions about this solution?

Here are some additional questions you might find interesting:

1. How can we generalize the method to find the intersection between any surface and a line?
2. How would you find the intersection between a line and a plane given in a different form, such as $Ax + By + Cz = D$?
3. What are other ways to represent lines in space, and how do they relate to parametric equations?
4. How can you find the distance between a point and a plane using a similar method?
5. What happens if there is no intersection? How can you determine this?

Tip: To solve similar problems, always start by converting the given line equation to parametric form and then substitute those parameters into the equation of the surface or plane to find the intersection point.