Given OA = a and OB = b, express the following vectors in terms of a and b only: (i) AB (ii) OD. Also, express OE in two ways and find k and h.

Here's a step-by-step solution to the problem: Part (a): Express the following vectors in terms of a and b only. a) (i) AB Given OA = a and OB = b. Using the triangle law of vector addition: AB = OB - OA AB = b - a The vector AB is b - a. a) (ii) OD First, find OC. We are given OB = (2)/(5)OC. Rearranging this, we get: OC = (5)/(2)OB Since OB = b: OC = (5)/(2)b Next, D is the midpoint of AC. Therefore, OD can be expressed as the average of OA and OC: OD = (1)/(2)(OA + OC) Substitute OA = a and OC = (5)/(2)b: OD = (1)/(2)(a + (5)/(2)b) OD = (1)/(2)a + (5)/(4)b The vector OD is (1)/(2)a + (5)/(4)b. Part (b): Express OE in two different ways, hence find the values of k and h. First expression for OE (using AE = kAB): We can express OE as OA + AE. OE = OA + AE Substitute OA = a and AE = kAB = k(b - a) (from part a(i)): OE = a + k(b - a) OE = a + kb - ka OE = (1 - k)a + kb (Equation 1) Second expression for OE (using OE = hOD): We are given OE = hOD. Substitute OD = (1)/(2)a + (5)/(4)b (from part a(ii)): OE = h((1)/(2)a + (5)/(4)b) OE = (h)/(2)a + (5h)/(4)b (Equation 2) Finding the values of k and h: Equate the two expressions for OE (Equation 1 and Equation 2): (1 - k)a + kb = (h)/(2)a + (5h)/(4)b Since a and b are non-parallel vectors, their coefficients must be equal. Equating coefficients of a: 1 - k = (h)/(2) (Equation 3) Equating coefficients of b: k = (5h)/(4) (Equation 4) Substitute Equation 4 into Equation 3: 1 - (5h)/(4) = (h)/(2) Multiply the entire equation by 4 to eliminate denominators: 4 - 5h = 2h 4 = 7h h = (4)/(7) Substitute the value of h back into Equation 4 to find k: k = (5)/(4) × (4)/(7) k = (5)/(7) The values are k = (5)/(7) and h = (4)/(7). Part (c): State the ratio in which E divides: c) (i) AB We found AE = kAB and k = (5)/(7). So, AE = (5)/(7)AB. This means that E divides AB in the ratio AE : EB. AE : EB = k : (1 - k) AE : EB = (5)/(7) : (1 - (5)/(7)) AE : EB = (5)/(7) : (2)/(7) AE : EB = 5 : 2 E divides AB in the ratio 5 : 2. c) (ii) OD We found OE = hOD and h = (4)/(7). So, $OE