Final answer:
To find the vector equation of a line, calculate the direction vector by subtracting the coordinates of one point from another, then add this vector to the position vector of one of the points, scaled by a parameter t.
Step-by-step explanation:
To find a vector equation for the line through two points, we must first determine the direction vector by subtracting the coordinates of the initial point from the terminal point. Then, we use one of the points as the point vector and combine it with the direction vector using a parameter. The vector equation of a line in two dimensions is r(t) = r0 + tv, where r0 is the position vector of a point on the line, v is the direction vector, and t is the parameter.
For example, if the two points are P1(x1, y1) and P2(x2, y2), the direction vector v would be calculated as follows:
- Identify the coordinates of P1 and P2.
- Subtract the coordinates of P1 from P2 to find the direction vector: v = <x2 - x1, y2 - y1>.
- Choose one of the points as our point vector, for example, r0 = <x1, y1>.
- Write the vector equation: r(t) = <x1, y1> + t<x2 - x1, y2 - y1>.