Results 1 to 4 of 4

Math Help - Position Vector of Reflection of a Point

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    2

    Question Position Vector of Reflection of a Point

    Hello, I am currently having trouble understanding how to find the reflection point of a point with the coordiantes of that point and the vector equation of the line. I have tried finding the maginitude of the foot of the prependicular, but I still could not get the answer. I hope my doubts can be cleared.

    This is the question :
    Find the position vector of the image C' of the point C(5,2,-1) under a reflection in the line l joining the points A(3,1,3) and B(-1,-1,-3).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570

    Vectors

    Hello MathJunction

    Welcome to Math Help Forum!
    Quote Originally Posted by MathJunction View Post
    Hello, I am currently having trouble understanding how to find the reflection point of a point with the coordiantes of that point and the vector equation of the line. I have tried finding the maginitude of the foot of the prependicular, but I still could not get the answer. I hope my doubts can be cleared.

    This is the question :
    Find the position vector of the image C' of the point C(5,2,-1) under a reflection in the line l joining the points A(3,1,3) and B(-1,-1,-3).
    Using vectors, suppose that the position vector of A is \vec{a} = 3\vec{i} + \vec{j} + 3\vec{k}. Similarly \vec{b} = -\vec{i} -\vec{j} -3\vec{k},\, \vec{c} = 5\vec{i} + 2\vec{j} -\vec{k}.

    Then if M is the mid-point of AB, its position vector is given by \vec{m} = \tfrac12(\vec{a}+\vec{b}) = -\vec{i}.

    \Rightarrow \vec{CM} = \vec{m} - \vec{c} = -6\vec{i} -2\vec{j} +\vec{k}

    Now if C' is the reflection of C in AB, \vec{MC'}=\vec{CM}

    \Rightarrow \vec{c'} = \vec{m} + \vec{MC'} =\vec{m} + \vec{CM}, where \vec{c'} is the position vector of C'.

    Can you complete it now?

    Grandad
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2009
    Posts
    2
    Quote Originally Posted by Grandad View Post
    Hello MathJunction

    Welcome to Math Help Forum!Using vectors, suppose that the position vector of A is \vec{a} = 3\vec{i} + \vec{j} + 3\vec{k}. Similarly \vec{b} = -\vec{i} -\vec{j} -3\vec{k},\, \vec{c} = 5\vec{i} + 2\vec{j} -\vec{k}.

    Then if M is the mid-point of AB, its position vector is given by \vec{m} = \tfrac12(\vec{a}+\vec{b}) = -\vec{i}.

    \Rightarrow \vec{CM} = \vec{m} - \vec{c} = -6\vec{i} -2\vec{j} +\vec{k}

    Now if C' is the reflection of C in AB, \vec{MC'}=\vec{CM}

    \Rightarrow \vec{c'} = \vec{m} + \vec{MC'} =\vec{m} + \vec{CM}, where \vec{c'} is the position vector of C'.

    Can you complete it now?

    Grandad
    Hello Grandad, thank you for taking some time off in helping me. I regret to tell you I am still blurred. Shouldn't vector m be 1/2(a+b) = (1,0,0) since vector AB is ( -4,-2,-6) ? Thus AM is half of AB which is (-2,-1,-3) and then OM would be vector AM + vector OA?

    Sorry to trouble you again.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570

    Vectors

    Hello MathJunction
    Quote Originally Posted by MathJunction View Post
    Hello Grandad, thank you for taking some time off in helping me. I regret to tell you I am still blurred. Shouldn't vector m be 1/2(a+b) = (1,0,0) since vector AB is ( -4,-2,-6) ? Thus AM is half of AB which is (-2,-1,-3) and then OM would be vector AM + vector OA?

    Sorry to trouble you again.
    You're quite right - I got a sign wrong. Sorry!

    \vec{m} = \tfrac12(\vec{a}+\vec{b})= \vec{i}

    So \vec{CM}=\vec{m}-\vec{c}=-4\vec{i} -2\vec{j} +\vec{k}

    Then continue as I said before.

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: January 26th 2011, 02:11 PM
  2. Replies: 0
    Last Post: January 25th 2011, 02:01 PM
  3. Reflection of a point
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 2nd 2010, 11:11 AM
  4. [SOLVED] Position vector of point of intersection of line and plane
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 1st 2010, 03:14 AM
  5. Reflection of Point onto Plane
    Posted in the Calculus Forum
    Replies: 13
    Last Post: September 27th 2009, 05:45 AM

Search Tags


/mathhelpforum @mathhelpforum