To find the vector equation and parametric equations for the line segment that joins the points P(0, 1, 1) and Q(12, 13, 14), we can follow these steps:
Step 1: Identify the Direction Vector
First, we need to determine the direction vector, which is obtained by subtracting the coordinates of point P from those of point Q:
Direction Vector = Q - P = (12 - 0, 13 - 1, 14 - 1) = (12, 12, 13)Step 2: Write the Vector Equation
The vector equation of the line segment can be expressed as:
R(t) = P + t(Q - P)Here, R(t) is the position vector at parameter t, where t varies from 0 to 1, P is the position vector of point P, and (Q – P) is the direction vector.
Substituting the points, we get:
R(t) = (0, 1, 1) + t(12, 12, 13)Step 3: Parametric Equations
Next, we can derive the parametric equations from the vector equation:
- x(t) = 0 + 12t = 12t
- y(t) = 1 + 12t
- z(t) = 1 + 13t
Now, we specify the parameter t in the range [0, 1]:
- When t = 0, we are at point P(0, 1, 1)
- When t = 1, we reach point Q(12, 13, 14)
Final Result
Therefore, the vector equation for the line segment joining P and Q is:
R(t) = (0, 1, 1) + t(12, 12, 13)And the parametric equations are:
- x(t) = 12t
- y(t) = 1 + 12t
- z(t) = 1 + 13t
Where t is in the interval [0, 1].