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?
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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!
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