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 is the solution to question 12: a) To show that vectors AB and CD are parallel, we need to express them in component form and show that one is a scalar multiple of the other. Step 1: Calculate vector AB. Given A(-2, 3) and B(-6, 1). AB = B - A = -6 \\ 1 - -2 \\ 3 = -6 - (-2) \\ 1 - 3 = -4 \\ -2 Step 2: Calculate vector CD. Given C(12, -1) and D(4, -5). CD = D - C = 4 \\ -5 - 12 \\ -1 = 4 - 12 \\ -5 - (-1) = -8 \\ -4 Step 3: Compare the two vectors. We observe that CD = -8 \\ -4 = 2 -4 \\ -2 = 2 AB. Since CD is a scalar multiple of AB (with scalar k=2), the vectors are parallel. Thus, AB and CD are indeed parallel. b) To find AB, we first need to find the coordinates of point A, which is the midpoint of MN. Step 1: Find the coordinates of point A. Given M(5, 6) and N(-1, 4). The midpoint formula is A = ( (x_1+x_2)/(2), (y_1+y_2)/(2) ). A = ( (5 + (-1))/(2), (6 + 4)/(2) ) = ( (4)/(2), (10)/(2) ) = (2, 5) Step 2: Calculate vector AB. Given B(-1, 5) and A(2, 5). AB = B - A = -1 \\ 5 - 2 \\ 5 = -1 - 2 \\ 5 - 5 = -3 \\ 0 The vector AB is -3 \\ 0 . c) To find the vector f, we solve the given vector equation. Step 1: Isolate the term with f. 2f - 3 \\ 7 = 1 \\ 3 Add 3 \\ 7 to both sides: 2f = 1 \\ 3 + 3 \\ 7 2f = 1+3 \\ 3+7 2f = 4 \\ 10 Step 2: Solve for f. Divide both sides by 2: f = (1)/(2) 4 \\ 10 f = (4)/(2) \\ (10)/(2) f = 2 \\ 5 The vector f is 2 \\ 5 . Last free one today — make it count tomorrow, or type /upgrade for unlimited.