Questions: Find a function r(t) that describes the line passing through P(2,9,4) and Q(7,8,9).

Solution Steps

To find the parametric equation of a line passing through two points \( P(2, 9, 4) \) and \( Q(7, 8, 9) \), we can use the following approach:

1. Determine the direction vector \(\mathbf{d}\) by subtracting the coordinates of \(P\) from \(Q\).
2. Use the point \(P\) and the direction vector \(\mathbf{d}\) to write the parametric equations for the line.

To find the parametric equation of the line passing through points \( P(2, 9, 4) \) and \( Q(7, 8, 9) \), we first determine the direction vector \(\mathbf{d}\). The direction vector \(\mathbf{d}\) is given by: \[ \mathbf{d} = Q - P = (7 - 2, 8 - 9, 9 - 4) = (5, -1, 5) \]

Using the point \( P \) and the direction vector \(\mathbf{d}\), we can write the parametric equation of the line as: \[ \mathbf{r}(t) = P + t \mathbf{d} = (2, 9, 4) + t(5, -1, 5) \] This can be expanded to: \[ \mathbf{r}(t) = (2 + 5t, 9 - t, 4 + 5t) \]

Final Answer