Calculating the distance along a line that each point would intersect at - r

Question

I would like to fit a line through two points from a random distribution of points, then calculate the location along that line that each point intersects it orthogonally. I am not interested in the residual distance of each point from the line (points above/below the line are treated equally), I am only interested in calculating the location along the line of where that point would intersect (e.g. points at different distances from the line but at the same orthogonal location would have the same value). The data aren't connected to the line explicitly as the abline is drawn from the location of only 2 points, and so i can't extract these values in a classic residual type way. I don't think this is difficult, but I can't wrap by head around how to calculate it and it's really bugging me!
I have explored the dist2d function but that calculates the orthogonal distance of each point to the line. Is there a way to use that value to the then calculate the hypotenuse from the data point to some fixed constant point on the line, and then in turn calculate the adjacent distance from that constant? I would really appreciate any help!
#here is some example starter code here to visualise what I mean
#get random data
r = rnorm(100)
t = rnorm(100)
#bind and turn into a df
data = cbind(r,t)
data = as.data.frame(data)
head(data)
#plot
plot(data)
#want to draw abline between 2 points
#isolate points of interest
#here randomly select first two rows
d = data[c(1:2),]
head(d)
#calculate abline through selected points
lm = lm(t ~ r, d)
abline(lm)
#draw points to see which ones they cut through
points(d$r, d$t, bg = "red", pch = 21)

Answer 1

This code below works.
# Create dataframe
data = data.frame(x = rnorm(100), y = rnorm(100))
plot(data, xlim=c(-3, 3), ylim=c(-3, 3))
# Select two points
data$x1_red <- data[1,1]; data$y1_red <- data[1,2]; data$x2_red <- data[2,1]; data$y2_red <- data[2,2];
points(data$x1_red, data$y1_red, bg = "red", pch = 21); points(data$x2_red, data$y2_red, bg = "red", pch = 21);
# Show a red line where the points intersect
# Get its slope (m_red) and intercept (b_red)
data$m_red <- (data[2,2] - data[1,2]) / (data[2,1] - data[1,1])
data$b_red <- data$y1_red - data$m * data$x1_red
abline(data$b_red, data$m_red, col='red')
# Calculate the orthogonal slope
data$m_blue <- (-1/data$m_red)
abline(0, data$m_blue, col='blue')
# Solve for each point's b-intercept (if using the blue slope)
# y = m_blue * x + b
# b = y - m_blue * x
data$b <- data$y - data$m_blue * data$x
# Solve for where each point (using the m_blue slope) intersects the red line (x' and y')
# y' = m_blue * x' + b
# y' = m_red * x' + b_red
# Set those equations equal to each other and solve for x'
data$x_intersect <- (data$b_red - data$b) / (data$m_blue - data$m_red)
# Then solve for y'
data$y_intersect <- data$m_blue * data$x_intersect + data$b
# Calculate the distance between the point and where it intersects the red line
data$dist <- sqrt( (data$x - data$x_intersect)^2 + (data$y - data$y_intersect)^2 )