Given position vectors of M, N, R from the origin are m, n, and r respectively. R is on MN such that MR = 1/4 RN. Find an expression for r in terms of m and n.

Alright Almighty@Seer — let's do this. Here's the solution for the first problem involving position vectors. Problem 1: Position Vectors Given: • Position vectors of M, N, R from the origin are m, n, and r respectively. • R is a point on MN such that MR = (1)/(4) RN. • Find an expression for r in terms of m and n. Step 1: Express the vectors MR and RN in terms of position vectors. The vector from point A to point B is given by AB = OB - OA, where OA and OB are the position vectors of A and B from the origin. MR = OR - OM = r - m RN = ON - OR = n - r Step 2: Substitute these expressions into the given relationship. The problem states that MR = (1)/(4) RN. r - m = (1)/(4) (n - r) Step 3: Solve the equation for r. First, multiply both sides by 4 to eliminate the fraction: 4(r - m) = n - r Distribute the 4 on the left side: 4r - 4m = n - r Gather all terms involving r on one side and other terms on the other side: 4r + r = n + 4m Combine like terms: 5r = 4m + n Finally, divide by 5 to isolate r: r = 4m + n5 This can also be written as: r = (4)/(5)m + (1)/(5)n The expression for r in terms of m and n is: r = 4m + n5 Send me the next one 📸