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
Here are the calculations for the magnitudes of the vectors and their sum. Step 1: Find the magnitude of AB. The magnitude of a vector V = xi + yj + zk is given by |V| = sqrt(x^2 + y^2 + z^2). For AB = 3i + 4j - 2k: |AB| = sqrt(3^2 + 4^2 + (-2)^2) |AB| = sqrt(9 + 16 + 4) |AB| = sqrt(29) Step 2: Find the magnitude of BC. For BC = 4i + 3j + 5k: |BC| = sqrt(4^2 + 3^2 + 5^2) |BC| = sqrt(16 + 9 + 25) |BC| = sqrt(50) |BC| = 5sqrt(2) Step 3: Find the magnitude of PQ. For PQ = 2i + 3j + 6k: |PQ| = sqrt(2^2 + 3^2 + 6^2) |PQ| = sqrt(4 + 9 + 36) |PQ| = sqrt(49) |PQ| = 7 Step 4: Find the sum of the three vectors. Let S be the sum of the three vectors: S = AB + BC + PQ. S = (3i + 4j - 2k) + (4i + 3j + 5k) + (2i + 3j + 6k) Combine the i, j, and k components: S = (3+4+2)i + (4+3+3)j + (-2+5+6)k S = 9i + 10j + 9k Step 5: Find the magnitude of the sum of the three vectors. For S = 9i + 10j + 9k: |S| = sqrt(9^2 + 10^2 + 9^2) |S| = sqrt(81 + 100 + 81) |S| = sqrt(262) The magnitudes of the individual vectors are: |AB| = sqrt(29) |BC| = 5sqrt(2) |PQ| = 7 The sum of the three vectors is: S = 9i + 10j + 9k The magnitude of the sum of the three vectors is: |S| = sqrt(262) Send me the next one 📸