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?
Find the equation of a line which is perpendicular to a given vector and passing through a known point
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0Instead of Cross product why can't we use the Dot product.Because the Dot product of any two perpendicular vectors are equal to 0. – 2012-05-13
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310k views? How??? – 2014-04-29
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015k views and 3 upvotes :( – 2015-04-29
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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
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!
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0Thanks...It helped me a lot. – 2012-05-13
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
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1Perhaps using MathJAX to format your mathematical expressions would help, such as \$ai+bj+ck\$ to produce $ai+bj+ck$. – 2014-04-29
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0The answer is not quite correct...because in 3-space, the thing perpendicular to a line L is a plane. So any line in that plane is perpendicular to the givenline. – 2014-04-29
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0That 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