1. Vector Equations of Lines Dr. Shildneck

2. Vector Definition of Line P l

3. P (x 1, y 1 ) l X (x, y)

4. Vector Definition of Line P (x 1, y 1 ) l X (x, y) O

5. Vector Definition of Line P (x 1, y 1 ) X (x, y) O

6. Another Way to Think About It… Let v be a vector in standard position parallel to the line through P(x 1, y 1 ) and X(x, y). Then dv is a vector of any size (depending on scalar d); dv could also be the opposite direction (if d is negative). P (x 1, y 1 ) l X (x, y) O

7. Another Way to Think About It… Now take dv (which encompasses ALL possible vectors parallel to v, and thus parallel to the line PQ) and reposition it to have the initial point P. To do so, mathematically, we have added the position vector to dv. P (x 1, y 1 ) l X (x, y) O This, then, gives us the equation of our line. The equation is based on our parallel vector, v (and all of it’s variations - based on d) and Its repositioning with a point on the line. +

8. EXAMPLE 1 Write the vector equation of a line through A(-1, 3) that is parallel to the vector v =.

9. EXAMPLE 2 Write the vector equation of a line that passes through A(1, 2) and B(-2, 5).

10. EXAMPLE 3 Find the point that is 60% of the way from the point A(1, 2) to the point B(-2, 5). Use vectors.