12. Lines in space (page 2)

12. Lines in space


The parametric scalar equations	Page 2 of 3

Suppose that the point P₀ has coordinates x₀,y₀,z₀, and the non-zero vector v has components a,b,c.

Let x,y,z be the coordinates of an arbitrary point P on the line l.

Then

v = ai + bj + ck

and the position vectors of P₀ and P are given by

----> OP0 = r0 = x0i + y0j + z0k

and

---> OP = r = xi + yj + zk

respectively. The parametric vector equation may be rewritten as

(x - x0)i + (y- y0)j + (z - z0)k = t(ai + bj + ck),

that is,

x - x0 = ta, y- y = tb, 0 z- z0 = tc,

with t R. These equations are called parametric scalar equations of the line.