Coordinate systems in R Programming - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of coordinate transformation using base R

What is the output of this R code that converts Cartesian coordinates (x, y) to polar coordinates (r, theta)?

R Programming

cartesian_to_polar <- function(x, y) {
  r <- sqrt(x^2 + y^2)
  theta <- atan2(y, x)
  return(c(r = r, theta = theta))
}

result <- cartesian_to_polar(3, 4)
print(round(result, 2))

A r 7.00, theta 0.93

B r 5.00, theta 0.93

C r 5.00, theta 0.64

D r 7.00, theta 0.64

💡 Hint

Recall that r is the distance from origin and theta is the angle in radians from the x-axis.

The distance r is sqrt(3^2 + 4^2) = 5. The angle theta is atan2(4, 3) ≈ 0.93 radians.

🧠 Conceptual

intermediate

Understanding coordinate system origin shifts

If you shift the origin of a 2D Cartesian coordinate system by vector (2, -3), what are the new coordinates of point (5, 7)?

A (3, 10)

B (7, 4)

C (3, 4)

D (7, 10)

💡 Hint

Subtract the shift vector from the original point coordinates.

Shifting the origin by (2, -3) places the new origin at old (2, -3). New coordinates = (5 - 2, 7 - (-3)) = (3, 10).

🔧 Debug

advanced

Identify the error in coordinate rotation code

What error does this R code produce when rotating a point (x, y) by angle theta (in radians)?

R Programming

rotate_point <- function(x, y, theta) {
  x_new <- x * cos(theta) - y * sin(theta)
  y_new <- x * sin(theta) + y * cos(theta)
  return(c(x_new, y_new))
}

rotate_point(1, 0, pi / 2)

A Returns numeric vector c(1, 0) without rotation

B Error: object 'theta' not found

C Returns numeric vector c(0, 1) as expected

D Error: non-numeric argument to binary operator

💡 Hint

Check if all variables are defined and used correctly.

The function correctly computes the rotated coordinates. Rotating (1,0) by 90 degrees (pi/2 radians) results in (0,1).

📝 Syntax

advanced

Syntax error in coordinate transformation function

Which option contains the correct syntax for defining a function that converts polar to Cartesian coordinates in R?

A

polar_to_cartesian <- function(r, theta) {
  x <- r * cos(theta)
  y <- r * sin(theta)
  return(c(x = x, y = y))
}

B

polar_to_cartesian <- function(r, theta) {
  x <- r * cos(theta)
  y <- r * sin(theta)
  c(x = x, y = y)
}

C

polar_to_cartesian <- function(r, theta) {
  x <- r * cos(theta)
  y <- r * sin(theta)
  return(x, y)
}

D

polar_to_cartesian <- function(r, theta) {
  x = r * cos(theta)
  y = r * sin(theta)
  return(c(x, y))
}

💡 Hint

Check the return statement syntax and how multiple values are returned.

Option A uses return() with a named vector correctly. Option A returns unnamed vector but is syntactically valid. Option A tries to return two values separately which is invalid. Option A returns vector without return() but is valid in R as last expression is returned.

🚀 Application

expert

Calculate distance between two points in different coordinate systems

Given point A in Cartesian coordinates (3, 4) and point B in polar coordinates (r = 5, theta = pi/3), what is the Euclidean distance between A and B?

A 3.0

B 4.0

C 5.0

D 0.6

💡 Hint

Convert point B to Cartesian coordinates first, then use distance formula.

Point B in Cartesian: x = 5*cos(pi/3) = 2.5, y = 5*sin(pi/3) ≈ 4.33. Distance = sqrt((3-2.5)^2 + (4-4.33)^2) ≈ sqrt(0.25 + 0.11) ≈ 0.6.