Advanced Math/Linear algebra 1
I have been having difficulty with this problem. Any help would be appreciated.
Consider a line L which goes by a point Q and is parallel to a vector v. Let P be any point and let R be its mirror image across the line L.
(a) Find a formula for R starting from P, Q and vector v. Sketch your construction.
(b) Assume that P(1, 6, 2), Q(-4, 1, 2) and vector v = <-2,-1,5>. Find R.
a) To start, let me denote vectors with a prime ', so that the vector V in the figure is written V' = (v1, v2, v3). The vector from point P to point Q is written PQ', etc.
The way I solved this is to recognize that the perpindicular vector R' from the vector V' to R is minus the perpindicular vector P' from V' to P. This is easy to see in the figure but trickier to write out in vector component form (especially for random orientations in 3-D). If we can describe P' in component form then R' is obtained by putting a minus sign in front of the components of P'. This may not be the most straightforward way to solve it but it should at least give you something to work with.
Let the point where the line PQ intersects V' be V0. Then P' goes from V0 to P. We also have the vector V0' from Q to V0. By the rules of vector addition we can write
P' = PQ' - V0'
PQ' is obtained by subtracting the coordinate values (x, y, z values) of the point Q from the point P; PQ' = (p1-q1, p2-q2, p3-q3). If we have values for p1, q1, etc., then we can evaluate this expression (Part b). For V0', we need a magnitude and direction. The magnitude |V0| = scalar/inner/dot product of PQ' and V'
PQ'･V' = (pq1)v1 + (pq2)v2 + (pq3)v3 = a number (not a vector).
V0' points in the same direction as V' so we need the unit vector in that direction = V'/|V| = (v1, v2, v3)/sqrt(v1^2+v2^2+v3^2), so
V0' = (PQ'･V') V'/|V|
R' = -P' = -[PQ' - (PQ'･V') V'/|V|].
b) The vector components needed to evaluate the coordinates of R using the expression for R' are obtained from the coordinates of P, Q and V. From the above, we can write the components of the various vectors in R'.
(your notation of v = <-2, -1, 5>, with the brackets, leads me to interpret V' as starting at point Q as in the figure)
For instance, by substitution
PQ'･V' = (pq1)v1 + (pq2)v2 + (pq3)v3 = (p1-q1)v1 + (p2-q2)v2 + (p3-q3)v3 = ((1-(-4))(-2) + (6-1)(-1) + (2-2)5 = -15
(this value can be negative if the angle between the vectors is 3Pi/2 < angle < pi/2; you should check this).
The various other quantities can also be obtained this way to get R' = (r1, r2, r3). I'll let you do the arithmetic.
Hope this helps.