PS the four correlations were of course computed completely separately, by starting up R four times and running one of the following four blocks of code.
N = 10^7
e = matrix(nrow = 2, ncol = N, byrow = FALSE, data = scan("JoyVector.txt", nlines = N))
alpha = 0 * pi / 180
beta = 45 * pi / 180
a = c(cos(alpha), sin(alpha))
b = c(cos(beta), sin(beta))
mean(sign(colSums(e * a)) * -sign(colSums(e * b)))
N = 10^7
e = matrix(nrow = 2, ncol = N, byrow = FALSE, data = scan("JoyVector.txt", nlines = N))
alpha = 0 * pi / 180
beta = 135 * pi / 180
a = c(cos(alpha), sin(alpha))
b = c(cos(beta), sin(beta))
mean(sign(colSums(e * a)) * -sign(colSums(e * b)))
N = 10^7
e = matrix(nrow = 2, ncol = N, byrow = FALSE, data = scan("JoyVector.txt", nlines = N))
alpha = 90 * pi / 180
beta = 45 * pi / 180
a = c(cos(alpha), sin(alpha))
b = c(cos(beta), sin(beta))
mean(sign(colSums(e * a)) * -sign(colSums(e * b)))
N = 10^7
e = matrix(nrow = 2, ncol = N, byrow = FALSE, data = scan("JoyVector.txt", nlines = N))
alpha = 90 * pi / 180
beta = 135 * pi / 180
a = c(cos(alpha), sin(alpha))
b = c(cos(beta), sin(beta))
mean(sign(colSums(e * a)) * -sign(colSums(e * b)))