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
Hey Pretty, good to see you again. To show that points P, Q, and R are collinear, we need to demonstrate that two vectors formed by these points (e.g., PQ and QR) are parallel and share a common point. Parallel vectors are scalar multiples of each other. Step 1: Express the position vectors in column form. Given position vectors are: OP = 6i - 2j - 3k = 6 \\ -2 \\ -3 OQ = 12i - 5j - 6k = 12 \\ -5 \\ -6 Due to the blurriness in the image for OR, and the requirement to "show that P, Q and R are collinear", we assume the last component of OR is -4k for collinearity to hold. OR = 8i - 3j - 4k = 8 \\ -3 \\ -4 Step 2: Calculate the vectors PQ and QR. PQ = OQ - OP = 12 \\ -5 \\ -6 - 6 \\ -2 \\ -3 = 12-6 \\ -5-(-2) \\ -6-(-3) = 6 \\ -3 \\ -3 QR = OR - OQ = 8 \\ -3 \\ -4 - 12 \\ -5 \\ -6 = 8-12 \\ -3-(-5) \\ -4-(-6) = -4 \\ 2 \\ 2 Step 3: Check if PQ is a scalar multiple of QR. We look for a scalar such that PQ = QR. Comparing the components: For the i component: 6 = (-4) = (6)/(-4) = -(3)/(2) For the j component: -3 = (2) = -(3)/(2) For the k component: -3 = (2) = -(3)/(2) Since = -(3)/(2) is consistent for all components, we have PQ = -(3)/(2) QR. Step 4: Conclude collinearity. Since PQ is a scalar multiple of QR, the vectors are parallel. As they share a common point Q, the points P, Q, and R lie on the same straight line. Therefore, P, Q, and R are collinear points. Send me the next one 📸