"Give the equations of two lines that meet at the point $(2, -3, 5)$ and which meet at right angles, but do not use that point in either of the equations."
I am having a bad time with this one. I figured out the first line is the vector $(4,-6,10)$, that vector goes through the point, and then I was trying to find a perpendicular vector to $(2,-3,5)$. But if I do the rule of $u \cdot v = 0$, the lines are going to meet at $(0,0,0)$ and I want them to meet at $(2,-3,5)$, an help please?
The general equation of a line is given by $\mathbf r(t) = \mathbf r_0 + t\mathbf v$. In particular, when our vector space is $\Bbb R^3$, this equation is just $(x,y,z) = (x_0, y_0, z_0) + t(v_1, v_2, v_3)$.
Here $(x_0, y_0, z_0)$ is any point on your line, $(v_1, v_2, v_3)$ is any vector parallel to your line, and $t$ is your variable.
We know a point that is on both lines, don't we? It's $(2,-3,5)$.
Basically now all you have to do is find $2$ nonzero vectors which are perpendicular to each other to be the $\mathbf v$'s in your $2$ equations. I suggest just choosing one randomly and then using the definition of orthogonality to find the other one.
Definition: Two vectors, $\mathbf a$ and $\mathbf b$ are orthogonal if and only if $\mathbf a \cdot \mathbf b = \langle \mathbf a, \mathbf b\rangle =0$ (I'm not sure what notation you use for the dot product so I put both of the common ones).
So once you've chosen any vector, say $(1,1,1)$, you just need to plug that into this definition to find any other vector orthogonal to it: $(1,1,1)\cdot(a,b,c)=0$. Just find any nonzero solution to this.
Once you have those two vectors you just plug in. So your two equations will be $(x,y,z) = (2,-3,5) + t(u_1, u_2, u_3)$ and $(x,y,z)=(2,-3,5)+s(v_1, v_2, v_3)$ where $(u_1, u_2, u_3)$ and $(v_1, v_2, v_3)$ are your two orthogonal (perpendicular) vectors.
However, the question specifically asks that you not include the vector $(2,-3,5)$ in your equations. So basically, you just need one more point on each of these lines to replace the $(2,-3,5)$ with. So how would you find those two vectors?