# Thread: [SOLVED] Interpreting this question about line eqn perpendicular to plane?

1. ## [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 $\displaystyle r<sub>0</sub>$ is a vector, and A the point whose position vector relative to the origin is $\displaystyle r<sub>0</sub>$.

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 $\displaystyle r<sub>0</sub>$, v and c.

=) yay

2. Originally Posted by Ciocolatta
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 $\displaystyle r<sub>0</sub>$ is a vector, and A the point whose position vector relative to the origin is $\displaystyle r<sub>0</sub>$.

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 $\displaystyle r<sub>0</sub>$, v and c.

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

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

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

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

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

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

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

$\displaystyle \vec b \cdot \vec v = c~\implies~(\vec a + k \cdot \vec v) \cdot \vec v = c$

3. 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.

4. Originally Posted by Ciocolatta
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 . ...
Originally Posted by Ciocolatta
... "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 $\displaystyle \overrightarrow{AB} = \vec b - \vec a = (\vec a + k \cdot \vec v) - \vec a = k\cdot \vec v$

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

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