compute a vector which is perpendicular to another given vector (all are in 3D)

Question

I have a vector $v_1$ (suppose $v_1= \langle a_1, b_1,c_1\rangle$) and this $v_1$ passes through the point $(x_1,y_1,z_1)$. Now I need a second vector, $v_2$ which is perpendicular to $v_1$. Suppose that $v_2$ is passing through the second point $(x_2,y_2,z_2)$. However, my final goal is to find the intersection point of above two lines. So, could you help me to find the vector which is perpendicular to another given vector? Please anyone help me.

Answer 1

Ok, so your line $\ell$ has parametric equations $$ x=x_1+at,\qquad y=y_1+bt,\qquad z=z_1+ct. $$ If $Q=(x_2,y_2,z_2)$ is a point, the equation of the plane $\pi$ perpendicular to $\ell$ and passing through $Q$ has equation $$ a(x-x_2)+b(y-y_2)+c(z-z_2)=0. $$ Now just plug the former equations into the latter and solve for $t$.