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
Given the vectors: r_1 = 2i + 2j + k r_2 = 6i - 3j + 2k The question asks for "the Radius and the magnitude". In the context of vectors, "magnitude" refers to the length of a vector. "Radius" is not a standard term for operations on two arbitrary vectors. We will interpret "the magnitude" as the magnitudes of the given vectors r_1 and r_2. For "the Radius", we will interpret it as the distance between the two points represented by the position vectors r_1 and r_2, which is the magnitude of the difference vector |r_2 - r_1|. Step 1: Find the magnitude of r_1. The magnitude of a vector v = xi + yj + zk is given by |v| = sqrt(x^2 + y^2 + z^2). For r_1 = 2i + 2j + k: |r_1| = sqrt(2^2 + 2^2 + 1^2) |r_1| = sqrt(4 + 4 + 1) |r_1| = sqrt(9) |r_1| = 3 The magnitude of r_1 is 3. Step 2: Find the magnitude of r_2. For r_2 = 6i - 3j + 2k: |r_2| = sqrt(6^2 + (-3)^2 + 2^2) |r_2| = sqrt(36 + 9 + 4) |r_2| = sqrt(49) |r_2| = 7 The magnitude of r_2 is 7. Step 3: Find the "Radius" (interpreted as the distance between the points represented by r_1 and r_2). First, calculate the difference vector r_2 - r_1: r_2 - r_1 = (6i - 3j + 2k) - (2i + 2j + k) r_2 - r_1 = (6-2)i + (-3-2)j + (2-1)k r_2 - r_1 = 4i - 5j + k Now, find the magnitude of this difference vector: |r_2 - r_1| = sqrt(4^2 + (-5)^2 + 1^2) |r_2 - r_1| = sqrt(16 + 25 + 1) |r_2 - r_1| = sqrt(42) The "Radius" (distance between the points) is sqrt(42).