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.
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Here's the solution to question 9: a) (i) Find MO in terms of p and/or r. Step 1: Express OM using known vectors. Since O P Q R is a rectangle, OR = r and RQ = OP = p. M is the midpoint of RQ, so RM = (1)/(2)RQ = (1)/(2)p. We can write OM = OR + RM. OM = r + (1)/(2)p Step 2: Find MO. MO is the negative of OM. MO = -OM = - ( r + (1)/(2)p ) MO = -r - (1)/(2)p (ii) Find MT in terms of p and/or r. Step 1: Express MT using known vectors. We can write MT = MQ + QT. M is the midpoint of RQ, so MQ = (1)/(2)RQ = (1)/(2)p. Given PT : TQ = 2 : 1, this means QT = (1)/(3)QP. Since O P Q R is a rectangle, QP = -PQ = -OR = -r. So, QT = (1)/(3)(-r) = -(1)/(3)r. Step 2: Substitute the expressions for MQ and QT. MT = (1)/(2)p + (-(1)/(3)r) MT = (1)/(2)p - (1)/(3)r (iii) Find OT in terms of p and/or r. Step 1: Express OT using known vectors. We can write OT = OP + PT. Given OP = p. Given PT : TQ = 2 : 1, this means PT = (2)/(3)PQ. Since O P Q R is a rectangle, PQ = OR = r. So, PT = (2)/(3)r. Step 2: Substitute the expressions for OP and PT. OT = p + (2)/(3)r OT = p + (2)/(3)r (iv) Given MT = 2k \\ -k and |MT| = sqrt(180). Find the positive value of k. Step 1: Use the formula for the magnitude of a vector. The magnitude of a vector x \\ y is sqrt(x^2 + y^2). For MT = 2k \\ -k , its magnitude is: |MT| = sqrt((2k)^2 + (-k)^2) |MT| = sqrt(4k^2 + k^2) |MT| = sqrt(5k^2) Step 2: Set the magnitude equal to the given value and solve for k. We are given |MT| = sqrt(180). sqrt(5k^2) = sqrt(180) Square both sides of the equation: 5k^2 = 180 k^2 = (180)/(5) k^2 = 36 Take the square root of both sides: k = ±sqrt(36) k = ± 6 The question asks for the positive value of k. k = 6 Last free one today — make it count tomorrow, or type /upgrade for unlimited.