Answer to Question #97318 in Geometry for Max

Question

Find the reflection of the point A(3, 1) in the line L with equation r · (2i − j) = 0.
Use your answer to find the reflection in L of the line r · (i − j) − 2 = 0.

Answer 1

Reflection of a Point

We are going to fins the reflection of a point in the a Line

Solution:

Given, Point A (3, 1)

Let the Equation of r as

$$\vec r = x \vec I + y \vec j$$

Equation of L is given as

$$\vec r.(2 \vec i - \vec j ) = 0\\ (x \vec i + y \vec j). (2 \vec i - \vec j ) =0$$

$$2x - y = 0$$

$$(since \space \vec i . \vec i = \vec j . \vec j = \vec k .\vec k = 1\space and \space \space \vec i . \vec j = \vec j . \vec k = \vec k . \vec i =0)$$

The reflection of point A(3, 1) lies on the  line perpendicular to the line $2x - y = 0$

Perpendicular line is

$$x +2 y + k= 0$$

The Point (3, 1) lies on this line

$$3 +2( 1) + k = 0 \\ k= -5$$

Perpendicular line is

$$x + 2y -5= 0$$

The Intersecting Point of the Lines 2x−y=0 and x+2y−5=0 is the midpoint of the points (3, 1) and Reflecting point

Plug $y = 2x$ in the equation $x + 2y - 5 = 0$

then

$$x + 4x - 5 = 0\\ x= 1 \space and \space y =2$$

Let ( m, n ) is the reflection point

$so,$

$$\frac {3+m} {2} = 1 \space and \space \frac {1+n} {2} = 2 \\ m = -1 \space and \space n = 3$$

Reflection Point is ( -1, 3)

(b).

Let

$$L_1 : \vec r .(\vec i - \vec j ) - 2=0\\ (x \vec i + y \vec j) .(\vec i - \vec j ) - 2 = 0\\ x - y - 2 = 0$$

The intersecting point of these two line is (- 2, - 4)

Consider a point on the Line L₁ as (2, 0)

Now, Let the image of the line L₁ about the line L is L₂

So, the image of the point (2, 0) will be lie on the line L₂

Let the image of the point (2, 0) is (p, q)

So, $(\frac { p+2} {2}, \frac {q }{2} )$ lies on the line L

$$2 (\frac { p+2} {2} ) - (\frac {q }{2}) = 0$$

$$2p + 4 = q$$

Slope of (p, q) and (2, 0) is

$$\frac {q} {p - 2} = \frac {-1} {2}$$

$$p + 2q =2$$

Solving the equations $2p + 4 = q \space and \space p + 2q = 2$

We get,

$$p = \frac {-6}{5} \space and \space q = \frac {8}{5}$$

Reflection in L Passes through ( -2, -4) and ($\frac {-6}{5}, \frac {8}{5}$ )

$$y - y_1 = \frac {y_2 - y_1}{x_2 - x_1} (x - x_1) \\ y + 4 = \frac {\frac {8}{5} + 4} {\frac {-6} {5} +2} (x +2)$$

$$y + 4 = 7 (x +2)$$

$$Answer: y = 7x +10$$