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
Okay, faith, this is the last one for today. We need to find the sine of the angle between the two given vectors. The vectors are: a = 3i - 2j + k b = i + 2j - 3k The sine of the angle between two vectors a and b is given by the formula: = |a × b||a| |b| Step 1: Calculate the cross product a × b. a × b = i & j & k \\ 3 & -2 & 1 \\ 1 & 2 & -3 = i((-2)(-3) - (1)(2)) - j((3)(-3) - (1)(1)) + k((3)(2) - (-2)(1)) = i(6 - 2) - j(-9 - 1) + k(6 - (-2)) = 4i + 10j + 8k Step 2: Calculate the magnitude of the cross product |a × b|. |a × b| = sqrt(4^2 + 10^2 + 8^2) = sqrt(16 + 100 + 64) = sqrt(180) Simplify the radical: = sqrt(36 × 5) = 6sqrt(5) Step 3: Calculate the magnitudes of vectors a and b. |a| = sqrt(3^2 + (-2)^2 + 1^2) = sqrt(9 + 4 + 1) = sqrt(14) |b| = sqrt(1^2 + 2^2 + (-3)^2) = sqrt(1 + 4 + 9) = sqrt(14) Step 4: Substitute the magnitudes into the formula for . = 6sqrt(5)sqrt(14) × sqrt(14) = 6sqrt(5)14 Simplify the fraction: = 3sqrt(5)7 The sine of the angle between the vectors is: 3sqrt(5)7 Last free one today — make it count tomorrow, or type /upgrade for unlimited.