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 your question: Given the position vectors of two points X and Y: x = 2i - 3j y = -i + 2j Point M lies on the line segment XY such that the ratio of the magnitudes |XM| : |MY| is 8:2. This means M divides the line segment XY internally in the ratio m:n = 8:2. We can simplify the ratio 8:2 to 4:1. So, m=4 and n=1. Step 1: Apply the section formula for internal division. The position vector of a point M that divides the line segment joining points with position vectors x and y in the ratio n:m (where n corresponds to x and m corresponds to y) is given by: m = nx + myn+m In our case, the ratio is |XM| : |MY| = 4:1. This means M is closer to Y. So, the ratio is n=1 for x and m=4 for y. Step 2: Substitute the given position vectors and the ratio into the formula. m = 1 · (2i - 3j) + 4 · (-i + 2j)1+4 Step 3: Simplify the expression. m = 2i - 3j - 4i + 8j5 Combine the i components and the j components: m = (2-4)i + (-3+8)j5 m = -2i + 5j5 m = -(2)/(5)i + (5)/(5)j m = -(2)/(5)i + j The position vector of M is -(2)/(5)i + j. Last free one today — make it count tomorrow, or type /upgrade for unlimited.