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
Step 1: Write down the position vectors of points A, B, and C. The position vectors are: OA = 1 \\ -1 OB = 5 \\ -3 OC = 11 \\ -6 Step 2: Calculate the vector AB. To find the vector AB, subtract the position vector of A from the position vector of B: AB = OB - OA = 5 \\ -3 - 1 \\ -1 = 5-1 \\ -3-(-1) = 4 \\ -2 Step 3: Calculate the vector BC. To find the vector BC, subtract the position vector of B from the position vector of C: BC = OC - OB = 11 \\ -6 - 5 \\ -3 = 11-5 \\ -6-(-3) = 6 \\ -3 Step 4: Show that AB and BC are parallel. Two vectors are parallel if one is a scalar multiple of the other. Let's check if BC = AB for some scalar . 6 \\ -3 = 4 \\ -2 From the x-components: 6 = 4 = (6)/(4) = (3)/(2) From the y-components: -3 = -2 = (-3)/(-2) = (3)/(2) Since = (3)/(2) is consistent for both components, we have BC = (3)/(2) AB. Step 5: Conclude collinearity. Since BC is a scalar multiple of AB, the vectors are parallel. Both vectors share a common point B. Therefore, points A, B, and C lie on the same straight line, meaning they are collinear. That's 2 down. 3 left today — send the next one.