Find the equation of a line which is perpendicular to a given vector and passing through a known point

There is given a vector $2 \vec i + \vec j - 3 \vec k$ and now I want to find the equation of a line that is perpendicular to the given vector and passing through a known point $(1,1,1)$. How can I solve this?

• Instead of Cross product why can't we use the Dot product.Because the Dot product of any two perpendicular vectors are equal to 0. Commented May 13, 2012 at 11:54
• 10k views? How??? Commented Apr 29, 2014 at 18:41
• 15k views and 3 upvotes :( Commented Apr 29, 2015 at 17:04
• @AlexM. Not a duplicate. Perpendicular to a plane is relatively straightforward, perpendicular to a vector in 3D is a tricky question indeed.
– user147263
Commented Feb 1, 2016 at 20:24

So, you are given the vector $(2,1,-3)$. Let $(2k,k,-3k)$ be the orthogonal projection of $(1,1,1)$ on $(2,1,-3)$. Then, $(2k-1,k-1,-3k-1)$ and $(2,1,-3)$ are orthogonal, giving: $4k-2+k-1+9k+3=0$ i.e. $k=0$. So, $(0,0,0)$, the origin is the projection. Hence, the line contains the points $(0,0,0)$ and $(1,1,1)$, so its equation is $x=y=z$, if my calculations are correct!
• Perhaps using MathJAX to format your mathematical expressions would help, such as \$ai+bj+ck\$ to produce $ai+bj+ck$. Commented Apr 29, 2014 at 19:15