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 magnitude and direction cosines for the given vector expressions. Given vectors: a = 3i - 2j + 4k b = 2i + j + 3k (i) For a - b Step 1: Calculate the vector a - b. a - b = (3i - 2j + 4k) - (2i + j + 3k) a - b = (3-2)i + (-2-1)j + (4-3)k a - b = i - 3j + k Step 2: Calculate the magnitude of a - b. |a - b| = sqrt((1)^2 + (-3)^2 + (1)^2) |a - b| = sqrt(1 + 9 + 1) |a - b| = sqrt(11) Step 3: Calculate the direction cosines of a - b. The direction cosines are (x)/(|r)|, (y)/(|r)|, (z)/(|r)|. = (1)/(sqrt(11)) = (-3)/(sqrt(11)) = (1)/(sqrt(11)) The magnitude is sqrt(11) and the direction cosines are ((1)/(sqrt(11)), (-3)/(sqrt(11)), (1)/(sqrt(11))). (ii) For 3a - 2b Step 1: Calculate 3a. 3a = 3(3i - 2j + 4k) = 9i - 6j + 12k Step 2: Calculate 2b. 2b = 2(2i + j + 3k) = 4i + 2j + 6k Step 3: Calculate the vector 3a - 2b. 3a - 2b = (9i - 6j + 12k) - (4i + 2j + 6k) 3a - 2b = (9-4)i + (-6-2)j + (12-6)k 3a - 2b = 5i - 8j + 6k Step 4: Calculate the magnitude of 3a - 2b. |3a - 2b| = sqrt((5)^2 + (-8)^2 + (6)^2) |3a - 2b| = sqrt(25 + 64 + 36) |3a - 2b| = sqrt(125) = sqrt(25 × 5) |3a - 2b| = 5sqrt(5) Step 5: Calculate the direction cosines of 3a - 2b. = (5)/(5sqrt(5)) = (1)/(sqrt(5)) = (-8)/(5sqrt(5)) = (6)/(5sqrt(5)) The magnitude is 5sqrt(5) and the direction cosines are ((1)/(sqrt(5)), (-8)/(5sqrt(5)), (6)/(5sqrt(5))). That's 2 down. 3 left today — send the next one.