Problem with two vector equations which is perpendicular to each other

Question

I am tackling a vector problem as shown below:

A line L1 has equation r=(-5,-3,2)+λ(-1,2,2)
A line L2 passing though the origin intersects L1, and is perpendicular to L1

Then here is the questions: (A)Find a vector equation of L2 (B)Deermine the shortest distance from the origin to L1

I don’t know how to find L2...the only thing I know is the dot product of L1 and L2 is equal to zero.Help !!

Answer 1

Hint: The equation of line $L_2$ is: $$ (0,0,0)+k(u,v,w) \ . $$

Edit: The $(0,0,0)$ is because the line $L_2$ goes through the origin. The generic vector $(u,v,w)$ is one that is orthogonal to the vector of $L_1$.

Edit 2: Since $L_1$ and $L_2$ are perpendicular, then the dot product of their vectors must be zero: $$ (u,v,w) \cdot (1, -1/2, -1/2) = 0 \ . $$

Edit 3: So, we have one equation on $u$, $v$ and $w$. You can obtain 3 more equations because $L_1$ and $L_2$ have to intercept themselves, one equation for the $x$ coordinate, another for $y$ and another for $z$. However, we have 5 unknowns: $\lambda$, $k$, $u$, $v$ and $w$. You'll find that you could choose either $u=1$ or $v=1$ or $w=1$ (choose one of those only). Now you have 4 equations with 4 unknowns.