7
$\begingroup$

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?

  • 1
    @AlexM. Not a duplicate. Perpendicular to a plane is relatively straightforward, perpendicular to a _vector_ in 3D is a tricky question indeed.2016-02-01

2 Answers 2

6

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!

  • 0
    Thanks...It helped me a lot.2012-05-13
-3

ai + b j + ck is the linear combination notation for a point in 3-space..It is meaningless to speak of a line being "perpendicular" to a point. ...What you want is the line perpendicular to the line L through (2 , 1 , -3) and the origin. The direction vector of L is (2,1,-3)..a vector orthogonal to that is (3,O 2)..so the formula of the line you want is of the form (1,1,1) + k(3,O,2)..k a real number.....which you may rewrite in cartesian notation

  • 0
    That would be a good comment to add to the answer that isn't correct. Please note that my comment only applies to how your answer is presented, not to its content. Note also that the question specifically asked for a line perpendicular to the given vector which passes through a given point, so it appears that the answer responds correctly to the question.2014-04-29