# Thread: Finding coordinate vector of point

1. ## Finding coordinate vector of point

A point P in $\mathbb{R}^n$ has coordinate vector $\mathbf{p}$. Find the coordinate vector of the point Q which is the reflection of P in the line $l$ which passes through the point $\mathbf{a}$ parallel to the direction $\mathbf{d}$

2. Originally Posted by acevipa
A point P in $\mathbb{R}^n$ has coordinate vector $\mathbf{p}$. Find the coordinate vector of the point Q which is the reflection of P in the line $l$ which passes through the point $\mathbf{a}$ parallel to the direction $\mathbf{d}$
I'm not certain if this will help:

1. Draw a sketch.

2. Since Q is the image of P by refelection about the line l the midpoint of $\overline{PQ}$ belongs to the line l and the vector $\vec d$ is perpendicular to $\overrightarrow{PQ}$. That means:

$\frac12(\vec p + \vec q)=\vec a + r \cdot \vec d$
and
$\vec d \cdot \overrightarrow{PQ} = 0$

3. Solve for $\vec q$.

I've got:

$\vec q = 2\vec a + 2r \cdot \vec d - \vec p$

3. Originally Posted by earboth
I'm not certain if this will help:

1. Draw a sketch.

2. Since Q is the image of P by refelection about the line l the midpoint of $\overline{PQ}$ belongs to the line l and the vector $\vec d$ is perpendicular to $\overrightarrow{PQ}$. That means:

$\frac12(\vec p + \vec q)=\vec a + r \cdot \vec d$
and
$\vec d \cdot \overrightarrow{PQ} = 0$

3. Solve for $\vec q$.

I've got:

$\vec q = 2\vec a + 2r \cdot \vec d - \vec p$
Thank you so much. That was really helpful