exercise with lines algebra

Question

This is an exercise I really can't solve by myself.

1) Let A(1,-1,0) a point on a line (e) line

2) Let (d) be a line perdicular to (e), given by a parametric equation.

How I can find the equation of (e)?

Answer 1

The parametric equations of a line in $\mathbb{R}$ are:

$$x=x_0+tu_1$$ $$y=y_0+tu_2$$ $$z=z_0+tu_3$$

Since A(1,-1,0) a point on the line (e) :

$$x=1+tu_1$$ $$y=-1+tu_2$$ $$z=tu_3$$

Since (d) is a line perdicular to (e) : ( $(d): (X,Y,Z)+t(v_1,v_2,v_3)$ )

$(u_1,u_2,u_3) \cdot (v_1,v_2,v_3)=0$