How does the reflection formula for vectors work?

In summary, the reflection formula for a vector v along an axis orthogonal to another vector a is v' = v - 2(va)a/(aa), where v and a are vectors and va and aa represent their dot product. This formula is derived by decomposing v into components parallel and perpendicular to a, and then reflecting the perpendicular component while reversing the parallel component. The magnitude of the reflected component is found by multiplying 2(va)/(aa) with the unit vector in the direction of a. The sign of the reflected component is determined by the position of v with respect to the reflection axis.
  • #1
Icosahedron
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Could someone explain how the reflection formula comes about?

R_a v = v - 2a(va)/ (aa) , where v and a are vectors and v is relected along the axis orthogonal to a

thanks
 
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  • #2
I presume that by "va" and "aa" you mean the dot product.

Decompose [itex]\vec{v}[/itex] into components parallel to and orthogonal to [itex]\vec{a}[/itex].

Recall that [itex]\vec{a}\cdot\vec{v}[/itex] can be defined as "[itex]|\vec{a}||\vec{v}| cos(\theta)[/itex] where [itex]\theta[/itex] is the angle between [itex]\vec{a}[/itex] and [itex]\vec{v}[/itex].

From simple trigonometry, the component of [itex]\vec{v}[/itex] parallel to [itex]\vec{a}[/itex] is [itex]|\vec{v}|cos(\theta)= \vec{a}\cdot\vec{v}/|\vec{a}|[/itex]. To get a "vector projection", a vector in the direction of [itex]\vec{a}[/itex] with that length, multiply by the unit vector in the direction of [itex]\vec{a}[/itex], [itex]\vec{a}/|\vec{a}|[/itex]. That gives [itex](\vec{a}\cdot\vec{v}/|\vec{a}|^2)\vec{a}[/itex] as the "vector component of [itex]\vec{v}[/itex] parallel to [itex]\vec{a}[/itex]". Since the components parallel and perpendicular to [itex]\vec{a}[/itex] must add to [itex]\vec{v}[/itex], the component perpendicular to [itex]\vec{a}[/itex] is [itex]\vec{v}-(\vec{a}\cdot\vec{v}/|\vec{a}|^2)\vec{a}[/itex].

Now, reflecting [itex]\vec{v}[/itex] along an axis orthogonal to [itex]\vec{a}[/itex] gives a new vector having the same component perpendicular to [itex]\vec{a}[/itex] but with the component parallel reversed:
[itex]\vec{v}-(\vec{a}\cdot\vec{v}/|\vec{a}|^2)\vec{a} -\vec{a}\cdot\vec{v}/|\vec{a}|^2)\vec{a}[/itex]
[itex]= \vec{v}-2(\vec{a}\cdot\vec{v}/|\vec{a}|^2)\vec{a}[/itex]
 
  • #3
I have an incomplete answer, but maybe you can finish it.

Let v' be the reflection of v over some axis. If you try to draw a vector s such that v = v' + s, using the triangle method to add the vectors v and s, you'll see that s is orthogonal to the reflection axis, and thus colinear to a.

The drawing should show an isosceles triangle, bisected by the reflection axis, where the two equally-sized sides are |v| and |v'|, the third side being of size |s|. From this triangle you should be able to calculate the magnitude of s as equal to 2 . |v| . cos theta, where "theta" is the angle between v (or v') and the normal to the reflection axis. (Try it on the drawing!)

Now, the expression 2 . |v| . cos theta is equal in magnitude to 2 . (va) / |a|. Multiplying this quantity by an unitary vector in the direction of a, namely a / |a|, you should get your vector s... or almost. There is a +/-1 factor not accounted yet for: has s the same direction of a or the opposite?

(This sign is the missing part that I was lazy to complete - I'd guess it has to do with v being or not on the same side of the axis than a; examining the sign of (va) should give a clue.)

Doing the multiplication above (and correcting the sign), you should get s = 2 (va) . a / |a|^2 = 2 (va) . a / (aa), and when replacing this expression for s into v' = v - s, you get your reflection formula.

Edit: Oh, I was pwned again. :)
 
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  • #4
thanks a lot!
 

1. What is a vector?

A vector is a mathematical quantity that has both magnitude and direction. It is typically represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow indicating the direction of the vector.

2. What is a reflection of a vector?

A reflection of a vector is the transformation of a vector across a line or plane. This results in the vector being flipped in direction, but maintaining the same magnitude.

3. How is a vector reflected across a line?

To reflect a vector across a line, we first draw a perpendicular line from the vector to the reflection line. Then, we measure the distance from the vector to the reflection line and create a new vector with the same magnitude, but in the opposite direction and the same distance from the reflection line.

4. What is the difference between a reflection and a rotation of a vector?

A reflection of a vector results in a mirror image of the original vector, while a rotation of a vector results in a change in the direction of the vector. Additionally, a reflection across a line will result in the same vector being reflected, while a rotation will result in a different vector.

5. Why are reflections of vectors important?

Reflections of vectors are important in many fields, including mathematics, physics, and computer graphics. They can be used to solve problems involving symmetry and to understand the behavior of light and sound waves. In computer graphics, reflections of vectors are used to create realistic images and animations.

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