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Math Help - [SOLVED] Interpreting this question about line eqn perpendicular to plane?

  1. #1
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    Red face [SOLVED] Interpreting this question about line eqn perpendicular to plane?

    Hi, I don't know how to approach this question, thanks in advance! =)

    Let v be a unit vector and c a real number, and let P be the plane whose equation is r.v = c . Suppose also that <b>r</b><sub>0</sub> is a vector, and A the point whose position vector relative to the origin is <b>r</b><sub>0</sub>.

    a) Find in parametric vector form, the equation of the line l that is perpendicular to P and passes through A.

    b) Let B be the point of intersection of the plane P and the line l in Part (a). Find a formula for the position vector of B relative to the origin, in t erms of <b>r</b><sub>0</sub>, v and c.

    =) yay
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  2. #2
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    Quote Originally Posted by Ciocolatta View Post
    Hi, I don't know how to approach this question, thanks in advance! =)

    Let v be a unit vector and c a real number, and let P be the plane whose equation is r.v = c . Suppose also that <b>r</b><sub>0</sub> is a vector, and A the point whose position vector relative to the origin is <b>r</b><sub>0</sub>.

    a) Find in parametric vector form, the equation of the line l that is perpendicular to P and passes through A.

    b) Let B be the point of intersection of the plane P and the line l in Part (a). Find a formula for the position vector of B relative to the origin, in t erms of <b>r</b><sub>0</sub>, v and c.

    =) yay
    1. According to the given equation of the plane the vector \vec v is the normal unit vector of P, that means \vec v \perp P

    2. Let \vec a be the position vector of A then the equation of the line passing through A and perpendicular to P has the equation:

    l: \vec r = \vec a+\lambda \cdot \vec v

    3. Let \vec b be the position vector of B.

    B is situated on the line l that means for a certain value \lambda = k you get:

    \vec b = \vec a + k \cdot \vec v

    4. Since B\in P the vector \vec b must satisfy the equation of the plane. Plug in the term for \vec b into the equation:

    \vec b \cdot \vec v = c~\implies~(\vec a + k \cdot \vec v) \cdot \vec v = c
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  3. #3
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    thank you!!!

    i was trying to do the next question but I'm not getting anywhere with it...

    it's "Use part b) to find a formula for the distance from A to P"

    I know it has to look something like d = |n.AB|/|n|

    I tried using the perpendicular line formula but I don't know how to derive it.
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  4. #4
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    Quote Originally Posted by Ciocolatta View Post
    Hi, I don't know how to approach this question, thanks in advance! =)

    Let v be a unit vector and c a real number, and let P be the plane whose equation is r.v = c . ...
    Quote Originally Posted by Ciocolatta View Post
    ... "Use part b) to find a formula for the distance from A to P"

    I know it has to look something like d = |n.AB|/|n|

    I tried using the perpendicular line formula but I don't know how to derive it.
    1. Calculate the vector \overrightarrow{AB} = \vec b - \vec a = (\vec a + k \cdot \vec v) - \vec a = k\cdot \vec v

    2. Since \vec v is a unit vector that means it's length is 1, you'll get:

    |\overrightarrow{AB}| = |k \cdot \vec v| = k
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