Question

How do you find the parametric equation for the line which pass by two points p1=(1,-2,1) p2=(0,-2,3)?

Answer 1

`mathbf x(lambda) = ((1), (-2),(1)) + lambda ((1), (0),(-2))`

Explanation:

Points `p_1=(1,-2,1)` and `p_2=(0,-2,3)` are connected by the direction vector, `vec d`, which is:

`vec d = p_1 - p_2 = (1,-2,1)^T - (0,-2,3)^T = ((1), (0),(-2))`

Any point on the line connecting the points can be found by starting at either of `p_1` or `p_2` and moving along the direction of `vec d`. IOW a generalised point with coordinates
`mathbf x = (x_1,x_2,x_3)^T` can be found according to the rule in vector form:

`((x_1(lambda)),(x_2(lambda)),(x_3(lambda))) = ((1), (-2),(1)) + lambda ((1), (0),(-2))`

ie `mathbf x(lambda) = ((1), (-2),(1)) + lambda ((1), (0),(-2))`

That's the parametric equation with `lambda` as the parameter.