This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the solution to the problem. a) Find vector LN. Step 1: Determine the position vectors OA and OB. Given the coordinates of A are (1, 6) and B are (15, 6), the position vectors from the origin O (0,0) are: OA = 1 \\ 6 OB = 15 \\ 6 Step 2: Determine the position vector ON. Point N is on OB such that 3ON = 2OB. This means ON = (2)/(3) OB. ON = (2)/(3) 15 \\ 6 = (2)/(3) × 15 \\ (2)/(3) × 6 = 10 \\ 4 Step 3: Determine the position vector OL. Line OA is produced to L such that OL = 3OA. This means OL = 3 OA. OL = 3 1 \\ 6 = 3 × 1 \\ 3 × 6 = 3 \\ 18 Step 4: Calculate vector LN. Vector LN can be found using the formula LN = ON - OL. LN = 10 \\ 4 - 3 \\ 18 = 10 - 3 \\ 4 - 18 = 7 \\ -14 The vector LN is 7 \\ -14 . b) Given that a point M is on LN such that LM:MN = 3:4, find the coordinate of M. Step 1: Use the section formula for position vectors. Point M divides the line segment LN in the ratio LM:MN = 3:4. This means M divides LN internally in the ratio m:n = 3:4. The position vector OM is given by the formula: OM = n OL + m ONm+n Substitute m=3, n=4, OL = 3 \\ 18 , and ON = 10 \\ 4 . OM = (4 3 \\ 18 + 3 10 \\ 4 )/(3+4) OM = ( 4 × 3 \\ 4 × 18 + 3 × 10 \\ 3 × 4 )/(7) OM = ( 12 \\ 72 + 30 \\ 12 )/(7) OM = ( 12 + 30 \\ 72 + 12 )/(7) OM = ( 42 \\ 84 )/(7) OM = (42)/(7) \\ (84)/(7) = 6 \\ 12 Step 2: State the coordinates of M. The position vector OM = 6 \\ 12 corresponds to the coordinates of M. The coordinate of M is (6, 12). 3 done, 2 left today. You're making progress.